Theorem rexxfr 4888

Description: Transfer existence from a variable x to another variable y contained in expression A. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.)

Hypotheses

Ref	Expression
ralxfr.1	|- ( y e. C -> A e. B )
ralxfr.2	|- ( x e. B -> E. y e. C x = A )
ralxfr.3	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
rexxfr	|- ( E. x e. B ph <-> E. y e. C ps )

Distinct variable groups:   ps, x   ph, y   x, A   x, y, B   x, C

Allowed substitution hints:   ph( x)   ps( y)   A( y)   C( y)

Proof of Theorem rexxfr

Step	Hyp	Ref	Expression
1		dfrex2 2996	. 2 |- ( E. x e. B ph <-> -. A. x e. B -. ph )
2		dfrex2 2996	. . 3 |- ( E. y e. C ps <-> -. A. y e. C -. ps )
3		ralxfr.1	. . . 4 |- ( y e. C -> A e. B )
4		ralxfr.2	. . . 4 |- ( x e. B -> E. y e. C x = A )
5		ralxfr.3	. . . . 5 |- ( x = A -> ( ph <-> ps ) )
6	5	notbid 308	. . . 4 |- ( x = A -> ( -. ph <-> -. ps ) )
7	3, 4, 6	ralxfr 4886	. . 3 |- ( A. x e. B -. ph <-> A. y e. C -. ps )
8	2, 7	xchbinxr 325	. 2 |- ( E. y e. C ps <-> -. A. x e. B -. ph )
9	1, 8	bitr4i 267	1 |- ( E. x e. B ph <-> E. y e. C ps )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202

This theorem is referenced by:  infm3  10982  reeff1o  24201  moxfr  37255