Theorem reuxfr3d 29329

Description: Transfer existential uniqueness from a variable x to another variable y contained in expression A. Cf. reuxfr2d 4891. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)

Hypotheses

Ref	Expression
reuxfr3d.1	|- ( ( ph /\ y e. C ) -> A e. B )
reuxfr3d.2	|- ( ( ph /\ x e. B ) -> E* y e. C x = A )

Assertion

Ref	Expression
reuxfr3d	|- ( ph -> ( E! x e. B E. y e. C ( x = A /\ ps ) <-> E! y e. C ps ) )

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

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

Proof of Theorem reuxfr3d

Step	Hyp	Ref	Expression
1		reuxfr3d.2	. . . . . . 7 |- ( ( ph /\ x e. B ) -> E* y e. C x = A )
2		rmoan 3406	. . . . . . 7 |- ( E* y e. C x = A -> E* y e. C ( ps /\ x = A ) )
3	1, 2	syl 17	. . . . . 6 |- ( ( ph /\ x e. B ) -> E* y e. C ( ps /\ x = A ) )
4		ancom 466	. . . . . . 7 |- ( ( ps /\ x = A ) <-> ( x = A /\ ps ) )
5	4	rmobii 3133	. . . . . 6 |- ( E* y e. C ( ps /\ x = A ) <-> E* y e. C ( x = A /\ ps ) )
6	3, 5	sylib 208	. . . . 5 |- ( ( ph /\ x e. B ) -> E* y e. C ( x = A /\ ps ) )
7	6	ralrimiva 2966	. . . 4 |- ( ph -> A. x e. B E* y e. C ( x = A /\ ps ) )
8		2reuswap 3410	. . . 4 |- ( A. x e. B E* y e. C ( x = A /\ ps ) -> ( E! x e. B E. y e. C ( x = A /\ ps ) -> E! y e. C E. x e. B ( x = A /\ ps ) ) )
9	7, 8	syl 17	. . 3 |- ( ph -> ( E! x e. B E. y e. C ( x = A /\ ps ) -> E! y e. C E. x e. B ( x = A /\ ps ) ) )
10		2reuswap2 29328	. . . 4 |- ( A. y e. C E* x ( x e. B /\ ( x = A /\ ps ) ) -> ( E! y e. C E. x e. B ( x = A /\ ps ) -> E! x e. B E. y e. C ( x = A /\ ps ) ) )
11		moeq 3382	. . . . . . 7 |- E* x x = A
12	11	moani 2525	. . . . . 6 |- E* x ( ( x e. B /\ ps ) /\ x = A )
13		ancom 466	. . . . . . . 8 |- ( ( ( x e. B /\ ps ) /\ x = A ) <-> ( x = A /\ ( x e. B /\ ps ) ) )
14		an12 838	. . . . . . . 8 |- ( ( x = A /\ ( x e. B /\ ps ) ) <-> ( x e. B /\ ( x = A /\ ps ) ) )
15	13, 14	bitri 264	. . . . . . 7 |- ( ( ( x e. B /\ ps ) /\ x = A ) <-> ( x e. B /\ ( x = A /\ ps ) ) )
16	15	mobii 2493	. . . . . 6 |- ( E* x ( ( x e. B /\ ps ) /\ x = A ) <-> E* x ( x e. B /\ ( x = A /\ ps ) ) )
17	12, 16	mpbi 220	. . . . 5 |- E* x ( x e. B /\ ( x = A /\ ps ) )
18	17	a1i 11	. . . 4 |- ( y e. C -> E* x ( x e. B /\ ( x = A /\ ps ) ) )
19	10, 18	mprg 2926	. . 3 |- ( E! y e. C E. x e. B ( x = A /\ ps ) -> E! x e. B E. y e. C ( x = A /\ ps ) )
20	9, 19	impbid1 215	. 2 |- ( ph -> ( E! x e. B E. y e. C ( x = A /\ ps ) <-> E! y e. C E. x e. B ( x = A /\ ps ) ) )
21		reuxfr3d.1	. . . 4 |- ( ( ph /\ y e. C ) -> A e. B )
22		biidd 252	. . . . 5 |- ( x = A -> ( ps <-> ps ) )
23	22	ceqsrexv 3336	. . . 4 |- ( A e. B -> ( E. x e. B ( x = A /\ ps ) <-> ps ) )
24	21, 23	syl 17	. . 3 |- ( ( ph /\ y e. C ) -> ( E. x e. B ( x = A /\ ps ) <-> ps ) )
25	24	reubidva 3125	. 2 |- ( ph -> ( E! y e. C E. x e. B ( x = A /\ ps ) <-> E! y e. C ps ) )
26	20, 25	bitrd 268	1 |- ( ph -> ( E! x e. B E. y e. C ( x = A /\ ps ) <-> E! y e. C ps ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E*wmo 2471   A.wral 2912   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:  reuxfr4d  29330