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

Axiom ax-7 1535
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. An alternate axiomatization could use ax467 1752 in place of ax-4 1692, ax-6o 1697, and ax-7 1535. (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 1532 . . 3  wff  A. y ph
4 vx . . 3  set  x
53, 4wal 1532 . 2  wff  A. x A. y ph
61, 4wal 1532 . . 3  wff  A. x ph
76, 2wal 1532 . 2  wff  A. y A. x ph
85, 7wi 6 1  wff  ( A. x A. y ph  ->  A. y A. x ph )
Colors of variables: wff set class
This axiom is referenced by:  a7s  1545  hbal  1567  alcom  1568  hbald  1614  nfald  1742  ax67  1749  ax467  1752  hbae  1841  hbae-o  1842  cbv1h  1871  sbal1  2089  ax11indalem  2113  ax11inda2ALT  2114  hbaltg  23519  pm11.71  26949  ax4567  26954  ax10ext  26959  hbalg  27358  hbalgVD  27715  hbexgVD  27716  hbae-x12  28260
  Copyright terms: Public domain W3C validator