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

Axiom ax-7 1749
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 ax7w 1733) 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 1549 . . 3  wff  A. y ph
4 vx . . 3  set  x
53, 4wal 1549 . 2  wff  A. x A. y ph
61, 4wal 1549 . . 3  wff  A. x ph
76, 2wal 1549 . 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  1750  hbal  1751  alcom  1752  hbald  1755  nfaldOLD  1872  cbv1hOLD  1975  hbae  2040  hbaeOLD  2041  sbal1  2202  hbae-o  2229  ax67  2241  ax467  2245  ax11indalem  2273  ax11inda2ALT  2274  hbaltg  25427  pm11.71  27564  ax4567  27569  ax10ext  27574  hbalg  28579  hbalgVD  28954  hbexgVD  28955
  Copyright terms: Public domain W3C validator