Axiom ax-7 1745

Description: Axiom of Quantifier Commutation. This axiom says universal quantifiers can be swapped. 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 1729) but is used as an auxiliary axiom to achieve metalogical completeness. (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 set y
3	1, 2	wal 1546	. . 3 wff A. y ph
4		vx	. . 3 set x
5	3, 4	wal 1546	. 2 wff A. x A. y ph
6	1, 4	wal 1546	. . 3 wff A. x ph
7	6, 2	wal 1546	. 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  1746  hbal  1747  alcom  1748  hbald  1751  nfaldOLD  1868  hbae  2005  cbv1h  2031  sbal1  2176  hbae-o  2203  ax67  2215  ax467  2219  ax11indalem  2247  ax11inda2ALT  2248  hbaltg  25378  pm11.71  27464  ax4567  27469  ax10ext  27474  hbalg  28353  hbalgVD  28726  hbexgVD  28727