Theorem ralxfr 4316

Description: Transfer universal quantification 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
ralxfr	|- ( A. x e. B ph <-> A. 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 ralxfr

Step	Hyp	Ref	Expression
1		ralxfr.1	. . . 4 |- ( y e. C -> A e. B )
2	1	adantl 272	. . 3 |- ( ( T. /\ y e. C ) -> A e. B )
3		ralxfr.2	. . . 4 |- ( x e. B -> E. y e. C x = A )
4	3	adantl 272	. . 3 |- ( ( T. /\ x e. B ) -> E. y e. C x = A )
5		ralxfr.3	. . . 4 |- ( x = A -> ( ph <-> ps ) )
6	5	adantl 272	. . 3 |- ( ( T. /\ x = A ) -> ( ph <-> ps ) )
7	2, 4, 6	ralxfrd 4312	. 2 |- ( T. -> ( A. x e. B ph <-> A. y e. C ps ) )
8	7	mptru 1305	1 |- ( A. x e. B ph <-> A. y e. C ps )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1296   T. wtru 1297   e. wcel 1445   A.wral 2370   E.wrex 2371

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635

This theorem is referenced by: (None)