Theorem reuxfr2 4892

Description: Transfer existential uniqueness from a variable x to another variable y contained in expression A. (Contributed by NM, 14-Nov-2004.) (Revised by NM, 16-Jun-2017.)

Hypotheses

Ref	Expression
reuxfr2.1	|- ( y e. B -> A e. B )
reuxfr2.2	|- ( x e. B -> E* y e. B x = A )

Assertion

Ref	Expression
reuxfr2	|- ( E! x e. B E. y e. B ( x = A /\ ph ) <-> E! y e. B ph )

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

Allowed substitution hints:   ph( y)   A( y)

Proof of Theorem reuxfr2

Step	Hyp	Ref	Expression
1		reuxfr2.1	. . . 4 |- ( y e. B -> A e. B )
2	1	adantl 482	. . 3 |- ( ( T. /\ y e. B ) -> A e. B )
3		reuxfr2.2	. . . 4 |- ( x e. B -> E* y e. B x = A )
4	3	adantl 482	. . 3 |- ( ( T. /\ x e. B ) -> E* y e. B x = A )
5	2, 4	reuxfr2d 4891	. 2 |- ( T. -> ( E! x e. B E. y e. B ( x = A /\ ph ) <-> E! y e. B ph ) )
6	5	trud 1493	1 |- ( E! x e. B E. y e. B ( x = A /\ ph ) <-> E! y e. B ph )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   T. wtru 1484   e. wcel 1990   E.wrex 2913   E!wreu 2914   E*wrmo 2915

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-v 3202

This theorem is referenced by: (None)