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Axiom ax-11 2171
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 2144) 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 1618 . . 3 wff 𝑦𝜑
4 vx . . 3 setvar 𝑥
53, 4wal 1618 . 2 wff 𝑥𝑦𝜑
61, 4wal 1618 . . 3 wff 𝑥𝜑
76, 2wal 1618 . 2 wff 𝑦𝑥𝜑
85, 7wi 4 1 wff (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
Colors of variables: wff setvar class
This axiom is referenced by:  alcoms  2172  hbal  2173  alcom  2174  hbald  2178  nfald  2298  hbae  2445  hbaltg  31989  bj-hbalt  32948  hbae-o  34661  axc711  34672  axc5c711  34676  ax12indalem  34703  ax12inda2ALT  34704  pm11.71  39068  axc5c4c711  39073  axc11next  39078  hbalg  39242  hbalgVD  39609  hbexgVD  39610
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