Theorem sb9i 1267

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 1179	. . . . 5 |- (A.y y = x -> ([y / y]ph <-> [y / x]ph))
2		drsb2 1234	. . . . 5 |- (A.y y = x -> ([y / y]ph <-> [x / y]ph))
3	1, 2	bitr3d 533	. . . 4 |- (A.y y = x -> ([y / x]ph <-> [x / y]ph))
4	3	dral1 1158	. . 3 |- (A.y y = x -> (A.y[y / x]ph <-> A.x[x / y]ph))
5	4	biimprd 154	. 2 |- (A.y y = x -> (A.x[x / y]ph -> A.y[y / x]ph))
6		hbsb2 1231	. . . . 5 |- (-. A.y y = x -> ([x / y]ph -> A.y[x / y]ph))
7	6	19.20ii 999	. . . 4 |- (A.x -. A.y y = x -> (A.x[x / y]ph -> A.xA.y[x / y]ph))
8	7	hbnaes 1152	. . 3 |- (-. A.y y = x -> (A.x[x / y]ph -> A.xA.y[x / y]ph))
9		stdpc4 1189	. . . . . 6 |- (A.x[x / y]ph -> [y / x][x / y]ph)
10		sbco 1256	. . . . . 6 |- ([y / x][x / y]ph <-> [y / x]ph)
11	9, 10	sylib 198	. . . . 5 |- (A.x[x / y]ph -> [y / x]ph)
12	11	19.20i 996	. . . 4 |- (A.yA.x[x / y]ph -> A.y[y / x]ph)
13	12	a7s 995	. . 3 |- (A.xA.y[x / y]ph -> A.y[y / x]ph)
14	8, 13	syl6 22	. 2 |- (-. A.y y = x -> (A.x[x / y]ph -> A.y[y / x]ph))
15	5, 14	pm2.61i 126	1 |- (A.x[x / y]ph -> A.y[y / x]ph)

Syntax hints:  -. wn 2   -> wi 3  A.wal 958  [wsbc 1174

This theorem is referenced by:  sb9 1268

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-9 969  ax-10 970  ax-12 972  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1127  ax-10o 1144  ax-11o 1222

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 985  df-sb 1176

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