MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax-7 Unicode version

Axiom ax-7 1708
Description: Axiom of Quantifier Commutation. This axiom says universal quantifiers can be swapped. One of the 4 axioms of pure predicate calculus. 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 1692) but is used as an auxiliary axiom to achieve metalogical completeness. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
ax-7  |-  ( A. x A. y ph  ->  A. y A. x ph )

Detailed syntax breakdown of Axiom ax-7
StepHypRef Expression
1 wph . . . 4  wff  ph
2 vy . . . 4  set  y
31, 2wal 1527 . . 3  wff  A. y ph
4 vx . . 3  set  x
53, 4wal 1527 . 2  wff  A. x A. y ph
61, 4wal 1527 . . 3  wff  A. x ph
76, 2wal 1527 . 2  wff  A. y A. x ph
85, 7wi 4 1  wff  ( A. x A. y ph  ->  A. y A. x ph )
Colors of variables: wff set class
This axiom is referenced by:  a7s  1709  hbal  1710  alcom  1711  hbald  1714  nfald  1775  hbae  1893  cbv1h  1918  sbal1  2065  hbae-o  2092  ax67  2104  ax467  2108  ax11indalem  2136  ax11inda2ALT  2137  hbaltg  24164  pm11.71  27596  ax4567  27601  ax10ext  27606  hbalg  28321  hbalgVD  28681  hbexgVD  28682  hbae-x12  29109
  Copyright terms: Public domain W3C validator