Theorem rexxfr 4503

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 4498	. 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 2167   E.wrex 2476

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765

This theorem is referenced by: (None)