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Axiom ax-7 1711
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 1695) 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 1529 . . 3  wff  A. y ph
4 vx . . 3  set  x
53, 4wal 1529 . 2  wff  A. x A. y ph
61, 4wal 1529 . . 3  wff  A. x ph
76, 2wal 1529 . 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  1712  hbal  1713  alcom  1714  hbald  1717  nfald  1779  hbae  1897  cbv1h  1923  sbal1  2070  hbae-o  2096  ax67  2108  ax467  2112  ax11indalem  2139  ax11inda2ALT  2140  hbaltg  23567  pm11.71  26997  ax4567  27002  ax10ext  27007  hbalg  27594  hbalgVD  27951  hbexgVD  27952  hbae-x12  28378
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