Theorem reuxfrd 4893

Description: Transfer existential uniqueness from a variable x to another variable y contained in expression A. Use reuhypd 4895 to eliminate the second hypothesis. (Contributed by NM, 16-Jan-2012.)

Hypotheses

Ref	Expression
reuxfrd.1	|- ( ( ph /\ y e. B ) -> A e. B )
reuxfrd.2	|- ( ( ph /\ x e. B ) -> E! y e. B x = A )
reuxfrd.3	|- ( x = A -> ( ps <-> ch ) )

Assertion

Ref	Expression
reuxfrd	|- ( ph -> ( E! x e. B ps <-> E! y e. B ch ) )

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

Allowed substitution hints:   ps( x)   ch( y)   A( y)

Proof of Theorem reuxfrd

Step	Hyp	Ref	Expression
1		reuxfrd.2	. . . . . 6 |- ( ( ph /\ x e. B ) -> E! y e. B x = A )
2		reurex 3160	. . . . . 6 |- ( E! y e. B x = A -> E. y e. B x = A )
3	1, 2	syl 17	. . . . 5 |- ( ( ph /\ x e. B ) -> E. y e. B x = A )
4	3	biantrurd 529	. . . 4 |- ( ( ph /\ x e. B ) -> ( ps <-> ( E. y e. B x = A /\ ps ) ) )
5		r19.41v 3089	. . . . 5 |- ( E. y e. B ( x = A /\ ps ) <-> ( E. y e. B x = A /\ ps ) )
6		reuxfrd.3	. . . . . . 7 |- ( x = A -> ( ps <-> ch ) )
7	6	pm5.32i 669	. . . . . 6 |- ( ( x = A /\ ps ) <-> ( x = A /\ ch ) )
8	7	rexbii 3041	. . . . 5 |- ( E. y e. B ( x = A /\ ps ) <-> E. y e. B ( x = A /\ ch ) )
9	5, 8	bitr3i 266	. . . 4 |- ( ( E. y e. B x = A /\ ps ) <-> E. y e. B ( x = A /\ ch ) )
10	4, 9	syl6bb 276	. . 3 |- ( ( ph /\ x e. B ) -> ( ps <-> E. y e. B ( x = A /\ ch ) ) )
11	10	reubidva 3125	. 2 |- ( ph -> ( E! x e. B ps <-> E! x e. B E. y e. B ( x = A /\ ch ) ) )
12		reuxfrd.1	. . 3 |- ( ( ph /\ y e. B ) -> A e. B )
13		reurmo 3161	. . . 4 |- ( E! y e. B x = A -> E* y e. B x = A )
14	1, 13	syl 17	. . 3 |- ( ( ph /\ x e. B ) -> E* y e. B x = A )
15	12, 14	reuxfr2d 4891	. 2 |- ( ph -> ( E! x e. B E. y e. B ( x = A /\ ch ) <-> E! y e. B ch ) )
16	11, 15	bitrd 268	1 |- ( ph -> ( E! x e. B ps <-> E! y e. B ch ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   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:  reuxfr  4894  riotaxfrd  6642