Theorem cbvralf 1799

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 groups:   z,A   x,y,z

Proof of Theorem cbvralf

Step	Hyp	Ref	Expression
1		ax-17 973	. . . . 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 1569	. . . 4 |- (x e. A -> A.y x e. A)
4		cbvralf.3	. . . 4 |- (ph -> A.yph)
5	3, 4	hbim 1009	. . 3 |- ((x e. A -> ph) -> A.y(x e. A -> ph))
6		ax-17 973	. . . . 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 1569	. . . 4 |- (y e. A -> A.x y e. A)
9		cbvralf.4	. . . 4 |- (ps -> A.xps)
10	8, 9	hbim 1009	. . 3 |- ((y e. A -> ps) -> A.x(y e. A -> ps))
11		eleq1 1537	. . . 4 |- (x = y -> (x e. A <-> y e. A))
12		cbvralf.5	. . . 4 |- (x = y -> (ph <-> ps))
13	11, 12	imbi12d 628	. . 3 |- (x = y -> ((x e. A -> ph) <-> (y e. A -> ps)))
14	5, 10, 13	cbval 1167	. 2 |- (A.x(x e. A -> ph) <-> A.y(y e. A -> ps))
15		df-ral 1652	. 2 |- (A.x e. A ph <-> A.x(x e. A -> ph))
16		df-ral 1652	. 2 |- (A.y e. A ps <-> A.y(y e. A -> ps))
17	14, 15, 16	3bitr4 183	1 |- (A.x e. A ph <-> A.y e. A ps)

Syntax hints:   -> wi 3   <-> wb 146  A.wal 956   = wceq 958   e. wcel 960  A.wral 1648

This theorem is referenced by:  cbvral 1801  ffnfvf 3835  hta 4738

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-cleq 1472  df-clel 1475  df-ral 1652

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