Theorem sb9i 2094

Description: Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.)

Assertion

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

Proof of Theorem sb9i

Step	Hyp	Ref	Expression
1		drsb1 2022	. . . . 5 |- (A.y y = x -> ([y / y]ph <-> [y / x]ph))
2		drsb2 2061	. . . . 5 |- (A.y y = x -> ([y / y]ph <-> [x / y]ph))
3	1, 2	bitr3d 246	. . . 4 |- (A.y y = x -> ([y / x]ph <-> [x / y]ph))
4	3	dral1 1965	. . 3 |- (A.y y = x -> (A.y[y / x]ph <-> A.x[x / y]ph))
5	4	biimprd 214	. 2 |- (A.y y = x -> (A.x[x / y]ph -> A.y[y / x]ph))
6		nfnae 1956	. . . 4 |- F/x -. A.y y = x
7		hbsb2 2057	. . . 4 |- (-. A.y y = x -> ([x / y]ph -> A.y[x / y]ph))
8	6, 7	alimd 1764	. . 3 |- (-. A.y y = x -> (A.x[x / y]ph -> A.xA.y[x / y]ph))
9		stdpc4 2024	. . . . . 6 |- (A.x[x / y]ph -> [y / x][x / y]ph)
10		sbco 2083	. . . . . 6 |- ([y / x][x / y]ph <-> [y / x]ph)
11	9, 10	sylib 188	. . . . 5 |- (A.x[x / y]ph -> [y / x]ph)
12	11	alimi 1559	. . . 4 |- (A.yA.x[x / y]ph -> A.y[y / x]ph)
13	12	a7s 1735	. . 3 |- (A.xA.y[x / y]ph -> A.y[y / x]ph)
14	8, 13	syl6 29	. 2 |- (-. A.y y = x -> (A.x[x / y]ph -> A.y[y / x]ph))
15	5, 14	pm2.61i 156	1 |- (A.x[x / y]ph -> A.y[y / x]ph)

Syntax hints:  -. wn 3   -> wi 4  A.wal 1540  [wsb 1648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649

This theorem is referenced by:  sb9  2095