Axiom ax-7 962

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 1023 in place of ax-4 973, ax-6o 978, and ax-7 962.

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 954	. . 3 wff A.yph
4		vx	. . 3 set x
5	3, 4	wal 954	. 2 wff A.xA.yph
6	1, 4	wal 954	. . 3 wff A.xph
7	6, 2	wal 954	. 2 wff A.yA.xph
8	5, 7	wi 3	1 wff (A.xA.yph -> A.yA.xph)

This axiom is referenced by:  a7s 991  hbal 1005  ax67 1020  ax467 1023  alcom 1032  hbald 1113  hbae 1145  cbv1 1162  sbal1 1346  ax11indalem 1368  ax11inda2ALT 1369  hbabd 1468

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