Theorem reuxfr4d 29330

Description: Transfer existential uniqueness from a variable x to another variable y contained in expression A. Cf. reuxfrd 4893. (Contributed by Thierry Arnoux, 7-Apr-2017.)

Hypotheses

Ref	Expression
reuxfr4d.1	|- ( ( ph /\ y e. C ) -> A e. B )
reuxfr4d.2	|- ( ( ph /\ x e. B ) -> E! y e. C x = A )
reuxfr4d.3	|- ( ( ph /\ x = A ) -> ( ps <-> ch ) )

Assertion

Ref	Expression
reuxfr4d	|- ( ph -> ( E! x e. B ps <-> E! y e. C ch ) )

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

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

Proof of Theorem reuxfr4d

Step	Hyp	Ref	Expression
1		reuxfr4d.2	. . . . . 6 |- ( ( ph /\ x e. B ) -> E! y e. C x = A )
2		reurex 3160	. . . . . 6 |- ( E! y e. C x = A -> E. y e. C x = A )
3	1, 2	syl 17	. . . . 5 |- ( ( ph /\ x e. B ) -> E. y e. C x = A )
4	3	biantrurd 529	. . . 4 |- ( ( ph /\ x e. B ) -> ( ps <-> ( E. y e. C x = A /\ ps ) ) )
5		r19.41v 3089	. . . . . 6 |- ( E. y e. C ( x = A /\ ps ) <-> ( E. y e. C x = A /\ ps ) )
6		reuxfr4d.3	. . . . . . . 8 |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )
7	6	pm5.32da 673	. . . . . . 7 |- ( ph -> ( ( x = A /\ ps ) <-> ( x = A /\ ch ) ) )
8	7	rexbidv 3052	. . . . . 6 |- ( ph -> ( E. y e. C ( x = A /\ ps ) <-> E. y e. C ( x = A /\ ch ) ) )
9	5, 8	syl5bbr 274	. . . . 5 |- ( ph -> ( ( E. y e. C x = A /\ ps ) <-> E. y e. C ( x = A /\ ch ) ) )
10	9	adantr 481	. . . 4 |- ( ( ph /\ x e. B ) -> ( ( E. y e. C x = A /\ ps ) <-> E. y e. C ( x = A /\ ch ) ) )
11	4, 10	bitrd 268	. . 3 |- ( ( ph /\ x e. B ) -> ( ps <-> E. y e. C ( x = A /\ ch ) ) )
12	11	reubidva 3125	. 2 |- ( ph -> ( E! x e. B ps <-> E! x e. B E. y e. C ( x = A /\ ch ) ) )
13		reuxfr4d.1	. . 3 |- ( ( ph /\ y e. C ) -> A e. B )
14		reurmo 3161	. . . 4 |- ( E! y e. C x = A -> E* y e. C x = A )
15	1, 14	syl 17	. . 3 |- ( ( ph /\ x e. B ) -> E* y e. C x = A )
16	13, 15	reuxfr3d 29329	. 2 |- ( ph -> ( E! x e. B E. y e. C ( x = A /\ ch ) <-> E! y e. C ch ) )
17	12, 16	bitrd 268	1 |- ( ph -> ( E! x e. B ps <-> E! y e. C 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:  rmoxfrdOLD  29332  rmoxfrd  29333  fcnvgreu  29472