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Axiom ax-7 1741
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 1725) 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 1546 . . 3  wff  A. y ph
4 vx . . 3  set  x
53, 4wal 1546 . 2  wff  A. x A. y ph
61, 4wal 1546 . . 3  wff  A. x ph
76, 2wal 1546 . 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  1742  hbal  1743  alcom  1744  hbald  1747  nfaldOLD  1865  hbae  1984  cbv1h  2004  sbal1  2153  hbae-o  2180  ax67  2192  ax467  2196  ax11indalem  2224  ax11inda2ALT  2225  hbaltg  25181  pm11.71  27258  ax4567  27263  ax10ext  27268  hbalg  27978  hbalgVD  28351  hbexgVD  28352  hbae-x12  29085
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