Theorem reusv1 4866

Description: Two ways to express single-valuedness of a class expression C ( y ). (Contributed by NM, 16-Dec-2012.) (Proof shortened by Mario Carneiro, 18-Nov-2016.) (Proof shortened by JJ, 7-Aug-2021.)

Assertion

Ref	Expression
reusv1	|- ( E. y e. B ph -> ( E! x e. A A. y e. B ( ph -> x = C ) <-> E. x e. A A. y e. B ( ph -> x = C ) ) )

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

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

Proof of Theorem reusv1

Step	Hyp	Ref	Expression
1		nfra1 2941	. . . 4 |- F/ y A. y e. B ( ph -> x = C )
2	1	nfmo 2487	. . 3 |- F/ y E* x A. y e. B ( ph -> x = C )
3		rsp 2929	. . . . . 6 |- ( A. y e. B ( ph -> x = C ) -> ( y e. B -> ( ph -> x = C ) ) )
4	3	com3l 89	. . . . 5 |- ( y e. B -> ( ph -> ( A. y e. B ( ph -> x = C ) -> x = C ) ) )
5	4	alrimdv 1857	. . . 4 |- ( y e. B -> ( ph -> A. x ( A. y e. B ( ph -> x = C ) -> x = C ) ) )
6		mo2icl 3385	. . . 4 |- ( A. x ( A. y e. B ( ph -> x = C ) -> x = C ) -> E* x A. y e. B ( ph -> x = C ) )
7	5, 6	syl6 35	. . 3 |- ( y e. B -> ( ph -> E* x A. y e. B ( ph -> x = C ) ) )
8	2, 7	rexlimi 3024	. 2 |- ( E. y e. B ph -> E* x A. y e. B ( ph -> x = C ) )
9		mormo 3158	. 2 |- ( E* x A. y e. B ( ph -> x = C ) -> E* x e. A A. y e. B ( ph -> x = C ) )
10		reu5 3159	. . 3 |- ( E! x e. A A. y e. B ( ph -> x = C ) <-> ( E. x e. A A. y e. B ( ph -> x = C ) /\ E* x e. A A. y e. B ( ph -> x = C ) ) )
11	10	rbaib 947	. 2 |- ( E* x e. A A. y e. B ( ph -> x = C ) -> ( E! x e. A A. y e. B ( ph -> x = C ) <-> E. x e. A A. y e. B ( ph -> x = C ) ) )
12	8, 9, 11	3syl 18	1 |- ( E. y e. B ph -> ( E! x e. A A. y e. B ( ph -> x = C ) <-> E. x e. A A. y e. B ( ph -> x = C ) ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   = 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:  cdleme25c  35643  cdleme29c  35664  cdlemefrs29cpre1  35686  cdlemk29-3  36199  cdlemkid5  36223  dihlsscpre  36523  mapdh9a  37079  mapdh9aOLDN  37080