MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax-11 Structured version   Visualization version   GIF version

Axiom ax-11 1971
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 1955) 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 1472 . . 3 wff 𝑦𝜑
4 vx . . 3 setvar 𝑥
53, 4wal 1472 . 2 wff 𝑥𝑦𝜑
61, 4wal 1472 . . 3 wff 𝑥𝜑
76, 2wal 1472 . 2 wff 𝑦𝑥𝜑
85, 7wi 4 1 wff (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
Colors of variables: wff setvar class
This axiom is referenced by:  alcoms  1972  hbal  1973  alcom  1974  hbald  1977  hbae  2207  hbaltg  30800  bj-hbalt  31693  hbae-o  33081  axc711  33092  axc5c711  33096  ax12indalem  33123  ax12inda2ALT  33124  pm11.71  37501  axc5c4c711  37506  axc11next  37511  hbalg  37674  hbalgVD  38045  hbexgVD  38046
  Copyright terms: Public domain W3C validator