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Axiom ax-11 2031
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 ax11w 2004) but is used as an auxiliary axiom scheme to achieve metalogical completeness. (Contributed by NM, 12-Mar-1993.)
Assertion
Ref Expression
ax-11 (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)

Detailed syntax breakdown of Axiom ax-11
StepHypRef Expression
1 wph . . . 4 wff 𝜑
2 vy . . . 4 setvar 𝑦
31, 2wal 1478 . . 3 wff 𝑦𝜑
4 vx . . 3 setvar 𝑥
53, 4wal 1478 . 2 wff 𝑥𝑦𝜑
61, 4wal 1478 . . 3 wff 𝑥𝜑
76, 2wal 1478 . 2 wff 𝑦𝑥𝜑
85, 7wi 4 1 wff (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
Colors of variables: wff setvar class
This axiom is referenced by:  alcoms  2032  hbal  2033  alcom  2034  hbald  2038  nfald  2162  hbae  2314  hbaltg  31411  bj-hbalt  32310  hbae-o  33665  axc711  33676  axc5c711  33680  ax12indalem  33707  ax12inda2ALT  33708  pm11.71  38076  axc5c4c711  38081  axc11next  38086  hbalg  38250  hbalgVD  38621  hbexgVD  38622
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