Theorem sb9i 1640

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		drsb1 1539	. . . . 5 |- (A.y y = x -> ([y / y]ph <-> [y / x]ph))
2		drsb2 1600	. . . . 5 |- (A.y y = x -> ([y / y]ph <-> [x / y]ph))
3	1, 2	bitr3d 589	. . . 4 |- (A.y y = x -> ([y / x]ph <-> [x / y]ph))
4	3	dral1 1515	. . 3 |- (A.y y = x -> (A.y[y / x]ph <-> A.x[x / y]ph))
5	4	biimprd 171	. 2 |- (A.y y = x -> (A.x[x / y]ph -> A.y[y / x]ph))
6		hbsb2 1597	. . . . 5 |- (-. A.y y = x -> ([x / y]ph -> A.y[x / y]ph))
7	6	al2imi 1341	. . . 4 |- (A.x -. A.y y = x -> (A.x[x / y]ph -> A.xA.y[x / y]ph))
8	7	hbnaes 1508	. . 3 |- (-. A.y y = x -> (A.x[x / y]ph -> A.xA.y[x / y]ph))
9		stdpc4 1550	. . . . . 6 |- (A.x[x / y]ph -> [y / x][x / y]ph)
10		sbco 1625	. . . . . 6 |- ([y / x][x / y]ph <-> [y / x]ph)
11	9, 10	sylib 215	. . . . 5 |- (A.x[x / y]ph -> [y / x]ph)
12	11	alimi 1338	. . . 4 |- (A.yA.x[x / y]ph -> A.y[y / x]ph)
13	12	a7s 1337	. . 3 |- (A.xA.y[x / y]ph -> A.y[y / x]ph)
14	8, 13	syl6 25	. 2 |- (-. A.y y = x -> (A.x[x / y]ph -> A.y[y / x]ph))
15	5, 14	pm2.61i 140	1 |- (A.x[x / y]ph -> A.y[y / x]ph)

Syntax hints:  -. wn 2   -> wi 3  A.wal 1296  [wsbc 1534

This theorem is referenced by:  sb9 1641

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-12 1310  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-11o 1588

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536

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