Axiom ax-7 964

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 1025 in place of ax-4 975, ax-6o 980, and ax-7 964.

Assertion

Ref	Expression
ax-7	|- (A.xA.yph -> A.yA.xph)

Detailed syntax breakdown of Axiom ax-7

Step	Hyp	Ref	Expression
1		wph	. . . 4 wff ph
2		vy	. . . 4 set y
3	1, 2	wal 956	. . 3 wff A.yph
4		vx	. . 3 set x
5	3, 4	wal 956	. 2 wff A.xA.yph
6	1, 4	wal 956	. . 3 wff A.xph
7	6, 2	wal 956	. 2 wff A.yA.xph
8	5, 7	wi 3	1 wff (A.xA.yph -> A.yA.xph)

This axiom is referenced by:  a7s 993  hbal 1007  ax67 1022  ax467 1025  alcom 1034  hbald 1115  hbae 1147  cbv1 1164  sbal1 1348  ax11indalem 1370  ax11inda2ALT 1371  hbabd 1471

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