Axiom ax-7 1458

Description: Axiom of Quantifier Commutation. This axiom says universal quantifiers can be swapped. One of the predicate logic axioms which do not involve equality. 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. (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

Step	Hyp	Ref	Expression
1		wph	. . . 4 wff ph
2		vy	. . . 4 setvar y
3	1, 2	wal 1361	. . 3 wff A. y ph
4		vx	. . 3 setvar x
5	3, 4	wal 1361	. 2 wff A. x A. y ph
6	1, 4	wal 1361	. . 3 wff A. x ph
7	6, 2	wal 1361	. 2 wff A. y A. x ph
8	5, 7	wi 4	1 wff ( A. x A. y ph -> A. y A. x ph )

This axiom is referenced by:  a7s  1464  alcoms  1486  hbal  1487  alcom  1488  hbald  1501  nfal  1586  hbae  1728  bj-hbalt  14811