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Axiom ax-7 1734
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 1718) but is used as an auxiliary axiom to achieve metalogical completeness. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
ax-7 (xyφyxφ)

Detailed syntax breakdown of Axiom ax-7
StepHypRef Expression
1 wph . . . 4 wff φ
2 vy . . . 4 setvar y
31, 2wal 1540 . . 3 wff yφ
4 vx . . 3 setvar x
53, 4wal 1540 . 2 wff xyφ
61, 4wal 1540 . . 3 wff xφ
76, 2wal 1540 . 2 wff yxφ
85, 7wi 4 1 wff (xyφyxφ)
Colors of variables: wff setvar class
This axiom is referenced by:  a7s  1735  hbal  1736  alcom  1737  hbald  1740  nfaldOLD  1853  hbae  1953  cbv1h  1978  sbal1  2126  hbae-o  2153  ax67  2165  ax467  2169  ax11indalem  2197  ax11inda2ALT  2198
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