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Axiom ax-11 2162
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 2135) 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 1536 . . 3 wff 𝑦𝜑
4 vx . . 3 setvar 𝑥
53, 4wal 1536 . 2 wff 𝑥𝑦𝜑
61, 4wal 1536 . . 3 wff 𝑥𝜑
76, 2wal 1536 . 2 wff 𝑦𝑥𝜑
85, 7wi 4 1 wff (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
Colors of variables: wff setvar class
This axiom is referenced by:  alcoms  2163  alcom  2164  hbal  2175  hbald  2176  hbsbw  2177  nfald  2349  hbae  2455  hbaltg  33109  bj-hbalt  34072  bj-nnflemaa  34150  bj-nfald  34497  hbae-o  36144  axc711  36155  axc5c711  36159  ax12indalem  36186  ax12inda2ALT  36187  pm11.71  41021  axc5c4c711  41025  axc11next  41030  hbalg  41181  hbalgVD  41531  hbexgVD  41532  ichal  43909
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