Theorem sb9i 920

Description: Commutation of quantification and substitution variables.

Assertion

Ref	Expression
sb9i	|- (A.x[x / y]ph -> A.y[y / x]ph)

Proof of Theorem sb9i

Step	Hyp	Ref	Expression
1		del43 856	. . . 4 |- (A.y y = x -> ([x / y]ph -> [x / x]ph))
2		del45 879	. . . 4 |- (A.y y = x -> ([x / x]ph -> [y / x]ph))
3	1, 2	syld 27	. . 3 |- (A.y y = x -> ([x / y]ph -> [y / x]ph))
4	3	del35 836	. 2 |- (A.y y = x -> (A.x[x / y]ph -> A.y[y / x]ph))
5		hbsb2 873	. . . . 5 |- (-. A.y y = x -> ([x / y]ph -> A.y[x / y]ph))
6	5	19.20ii 692	. . . 4 |- (A.x -. A.y y = x -> (A.x[x / y]ph -> A.xA.y[x / y]ph))
7	6	eq6s 827	. . 3 |- (-. A.y y = x -> (A.x[x / y]ph -> A.xA.y[x / y]ph))
8		stdpc4 869	. . . . . 6 |- (A.x[x / y]ph -> [y / x][x / y]ph)
9		sbco 910	. . . . . 6 |- ([y / x][x / y]ph <-> [y / x]ph)
10	8, 9	sylib 173	. . . . 5 |- (A.x[x / y]ph -> [y / x]ph)
11	10	19.20i 691	. . . 4 |- (A.yA.x[x / y]ph -> A.y[y / x]ph)
12	11	a7s 689	. . 3 |- (A.xA.y[x / y]ph -> A.y[y / x]ph)
13	7, 12	syl6 23	. 2 |- (-. A.y y = x -> (A.x[x / y]ph -> A.y[y / x]ph))
14	4, 13	pm2.61i 110	1 |- (A.x[x / y]ph -> A.y[y / x]ph)

Syntax hints:  -. wn 1   -> wi 2  A.wal 672   = weq 797  [wsb 852

This theorem is referenced by:  sb9 921

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853

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