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Axiom ax-7 1758
Description: Axiom of Quantifier Commutation. This axiom says universal quantifiers can be swapped. Axiom scheme C6' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Lemma 12 of [Monk2] p. 109 and Axiom C5-3 of [Monk2] p. 113. This axiom scheme is logically redundant (see ax7w 1742) but is used as an auxiliary axiom to achieve metalogical completeness. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
ax-7 (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)

Detailed syntax breakdown of Axiom ax-7
StepHypRef Expression
1 wph . . . 4 wff 𝜑
2 vy . . . 4 set 𝑦
31, 2wal 1556 . . 3 wff 𝑦𝜑
4 vx . . 3 set 𝑥
53, 4wal 1556 . 2 wff 𝑥𝑦𝜑
61, 4wal 1556 . . 3 wff 𝑥𝜑
76, 2wal 1556 . 2 wff 𝑦𝑥𝜑
85, 7wi 4 1 wff (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
Colors of variables: wff set class
This axiom is referenced by:  a7s  1759  hbal  1760  alcom  1761  hbald  1764  nfaldOLD  1883  cbv1hOLD  1990  hbae  2053  hbaeOLD  2054  sbal1OLD  2239  hbae-o  2271  ax67  2283  ax467  2287  ax11indalem  2314  ax11inda2ALT  2315  hbaltg  25892  pm11.71  28093  ax4567  28098  ax10ext  28103  hbalg  29608  hbalgVD  29985  hbexgVD  29986
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