Theorem cbvralf 1330

Description: Rule used to change bound variables with implicit substitution.

Hypotheses

Ref	Expression
cbvralf.1	|- (z e. A -> A.x z e. A)
cbvralf.2	|- (z e. A -> A.y z e. A)
cbvralf.3	|- (ph -> A.yph)
cbvralf.4	|- (ps -> A.xps)
cbvralf.5	|- (x = y -> (ph <-> ps))

Assertion

Ref	Expression
cbvralf	|- (A.x e. A ph <-> A.y e. A ps)

Distinct variable group(s):   z,A   x,y,z

Proof of Theorem cbvralf

Step	Hyp	Ref	Expression
1		ax-17 925	. . . . 5 |- (z e. x -> A.y z e. x)
2		cbvralf.2	. . . . 5 |- (z e. A -> A.y z e. A)
3	1, 2	hbel 1172	. . . 4 |- (x e. A -> A.y x e. A)
4		cbvralf.3	. . . 4 |- (ph -> A.yph)
5	3, 4	hbim 702	. . 3 |- ((x e. A -> ph) -> A.y(x e. A -> ph))
6		ax-17 925	. . . . 5 |- (z e. y -> A.x z e. y)
7		cbvralf.1	. . . . 5 |- (z e. A -> A.x z e. A)
8	6, 7	hbel 1172	. . . 4 |- (y e. A -> A.x y e. A)
9		cbvralf.4	. . . 4 |- (ps -> A.xps)
10	8, 9	hbim 702	. . 3 |- ((y e. A -> ps) -> A.x(y e. A -> ps))
11		eleq1 1149	. . . 4 |- (x = y -> (x e. A <-> y e. A))
12		cbvralf.5	. . . 4 |- (x = y -> (ph <-> ps))
13	11, 12	imbi12d 474	. . 3 |- (x = y -> ((x e. A -> ph) <-> (y e. A -> ps)))
14	5, 10, 13	cbval 848	. 2 |- (A.x(x e. A -> ph) <-> A.y(y e. A -> ps))
15		df-ral 1205	. 2 |- (A.x e. A ph <-> A.x(x e. A -> ph))
16		df-ral 1205	. 2 |- (A.y e. A ps <-> A.y(y e. A -> ps))
17	14, 15, 16	3bitr4 158	1 |- (A.x e. A ph <-> A.y e. A ps)

Syntax hints:   -> wi 2   <-> wb 127  A.wal 672   = weq 797   e. wel 803   e. wcel 1092  A.wral 1201

This theorem is referenced by:  cbvral 1331  hta 3619

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-12 802  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-cleq 1097  df-clel 1099  df-ral 1205

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