Theorem cbvrexf 1797

Description: Rule used to change bound variables with implicit substitution. (Contributed by FL, 27-Apr-2008.)

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
cbvrexf	|- (E.x e. A ph <-> E.y e. A ps)

Distinct variable groups:   z,A   x,y,z

Proof of Theorem cbvrexf

Step	Hyp	Ref	Expression
1		ax-17 971	. . . . 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 1566	. . . 4 |- (x e. A -> A.y x e. A)
4		cbvralf.3	. . . 4 |- (ph -> A.yph)
5	3, 4	hban 1009	. . 3 |- ((x e. A /\ ph) -> A.y(x e. A /\ ph))
6		ax-17 971	. . . . 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 1566	. . . 4 |- (y e. A -> A.x y e. A)
9		cbvralf.4	. . . 4 |- (ps -> A.xps)
10	8, 9	hban 1009	. . 3 |- ((y e. A /\ ps) -> A.x(y e. A /\ ps))
11		eleq1 1534	. . . 4 |- (x = y -> (x e. A <-> y e. A))
12		cbvralf.5	. . . 4 |- (x = y -> (ph <-> ps))
13	11, 12	anbi12d 628	. . 3 |- (x = y -> ((x e. A /\ ph) <-> (y e. A /\ ps)))
14	5, 10, 13	cbvex 1166	. 2 |- (E.x(x e. A /\ ph) <-> E.y(y e. A /\ ps))
15		df-rex 1650	. 2 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
16		df-rex 1650	. 2 |- (E.y e. A ps <-> E.y(y e. A /\ ps))
17	14, 15, 16	3bitr4 183	1 |- (E.x e. A ph <-> E.y e. A ps)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  E.wex 980  E.wrex 1646

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-cleq 1469  df-clel 1472  df-rex 1650

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