Axiom ax-7 954

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. An alternate axiomatization could use ax467 997 in place of ax-4 951, ax-6 953, and ax-7 954.

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 950	. . 3 wff A.yph
4		vx	. . 3 set x
5	3, 4	wal 950	. 2 wff A.xA.yph
6	1, 4	wal 950	. . 3 wff A.xph
7	6, 2	wal 950	. 2 wff A.yA.xph
8	5, 7	wi 3	1 wff (A.xA.yph -> A.yA.xph)

This axiom is referenced by:  a7s 967  hbal 981  ax67 994  ax467 997  alcom 1008  hbald 1089  hbae 1128  cbv1 1145  sbal1 1328  ax11indalem 1345  ax11inda2ALT 1346  hbabd 1445

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