Theorem reuan 41180

Description: Introduction of a conjunct into restricted uniqueness quantifier, analogous to euan 2530. (Contributed by Alexander van der Vekens, 2-Jul-2017.)

Hypothesis

Ref	Expression
rmoanim.1	|- F/ x ph

Assertion

Ref	Expression
reuan	|- ( E! x e. A ( ph /\ ps ) <-> ( ph /\ E! x e. A ps ) )

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   ps( x)

Proof of Theorem reuan

Step	Hyp	Ref	Expression
1		rmoanim.1	. . . . . 6 |- F/ x ph
2		simpl 473	. . . . . . 7 |- ( ( ph /\ ps ) -> ph )
3	2	a1i 11	. . . . . 6 |- ( x e. A -> ( ( ph /\ ps ) -> ph ) )
4	1, 3	rexlimi 3024	. . . . 5 |- ( E. x e. A ( ph /\ ps ) -> ph )
5	4	adantr 481	. . . 4 |- ( ( E. x e. A ( ph /\ ps ) /\ E* x e. A ( ph /\ ps ) ) -> ph )
6		simpr 477	. . . . . 6 |- ( ( ph /\ ps ) -> ps )
7	6	reximi 3011	. . . . 5 |- ( E. x e. A ( ph /\ ps ) -> E. x e. A ps )
8	7	adantr 481	. . . 4 |- ( ( E. x e. A ( ph /\ ps ) /\ E* x e. A ( ph /\ ps ) ) -> E. x e. A ps )
9		nfre1 3005	. . . . . 6 |- F/ x E. x e. A ( ph /\ ps )
10	4	adantr 481	. . . . . . . . 9 |- ( ( E. x e. A ( ph /\ ps ) /\ x e. A ) -> ph )
11	10	a1d 25	. . . . . . . 8 |- ( ( E. x e. A ( ph /\ ps ) /\ x e. A ) -> ( ps -> ph ) )
12	11	ancrd 577	. . . . . . 7 |- ( ( E. x e. A ( ph /\ ps ) /\ x e. A ) -> ( ps -> ( ph /\ ps ) ) )
13	6, 12	impbid2 216	. . . . . 6 |- ( ( E. x e. A ( ph /\ ps ) /\ x e. A ) -> ( ( ph /\ ps ) <-> ps ) )
14	9, 13	rmobida 3129	. . . . 5 |- ( E. x e. A ( ph /\ ps ) -> ( E* x e. A ( ph /\ ps ) <-> E* x e. A ps ) )
15	14	biimpa 501	. . . 4 |- ( ( E. x e. A ( ph /\ ps ) /\ E* x e. A ( ph /\ ps ) ) -> E* x e. A ps )
16	5, 8, 15	jca32 558	. . 3 |- ( ( E. x e. A ( ph /\ ps ) /\ E* x e. A ( ph /\ ps ) ) -> ( ph /\ ( E. x e. A ps /\ E* x e. A ps ) ) )
17		reu5 3159	. . 3 |- ( E! x e. A ( ph /\ ps ) <-> ( E. x e. A ( ph /\ ps ) /\ E* x e. A ( ph /\ ps ) ) )
18		reu5 3159	. . . 4 |- ( E! x e. A ps <-> ( E. x e. A ps /\ E* x e. A ps ) )
19	18	anbi2i 730	. . 3 |- ( ( ph /\ E! x e. A ps ) <-> ( ph /\ ( E. x e. A ps /\ E* x e. A ps ) ) )
20	16, 17, 19	3imtr4i 281	. 2 |- ( E! x e. A ( ph /\ ps ) -> ( ph /\ E! x e. A ps ) )
21		ibar 525	. . . . 5 |- ( ph -> ( ps <-> ( ph /\ ps ) ) )
22	21	adantr 481	. . . 4 |- ( ( ph /\ x e. A ) -> ( ps <-> ( ph /\ ps ) ) )
23	1, 22	reubida 3124	. . 3 |- ( ph -> ( E! x e. A ps <-> E! x e. A ( ph /\ ps ) ) )
24	23	biimpa 501	. 2 |- ( ( ph /\ E! x e. A ps ) -> E! x e. A ( ph /\ ps ) )
25	20, 24	impbii 199	1 |- ( E! x e. A ( ph /\ ps ) <-> ( ph /\ E! x e. A ps ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   F/wnf 1708   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-10 2019  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-eu 2474  df-mo 2475  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920

This theorem is referenced by:  2reu7  41191  2reu8  41192