Theorem rexxfr 4483

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		ralxfr.1	. . . 4 |- ( y e. C -> A e. B )
2	1	adantl 277	. . 3 |- ( ( T. /\ y e. C ) -> A e. B )
3		ralxfr.2	. . . 4 |- ( x e. B -> E. y e. C x = A )
4	3	adantl 277	. . 3 |- ( ( T. /\ x e. B ) -> E. y e. C x = A )
5		ralxfr.3	. . . 4 |- ( x = A -> ( ph <-> ps ) )
6	5	adantl 277	. . 3 |- ( ( T. /\ x = A ) -> ( ph <-> ps ) )
7	2, 4, 6	rexxfrd 4478	. 2 |- ( T. -> ( E. x e. B ph <-> E. y e. C ps ) )
8	7	mptru 1373	1 |- ( E. x e. B ph <-> E. y e. C ps )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   T. wtru 1365   e. wcel 2160   E.wrex 2469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754

This theorem is referenced by: (None)