Theorem eusvobj2 6643

Description: Specify the same property in two ways when class B ( y ) is single-valued. (Contributed by NM, 1-Nov-2010.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)

Hypothesis

Ref	Expression
eusvobj1.1	|- B e. _V

Assertion

Ref	Expression
eusvobj2	|- ( E! x E. y e. A x = B -> ( E. y e. A x = B <-> A. y e. A x = B ) )

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

Allowed substitution hint:   B( y)

Proof of Theorem eusvobj2

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		euabsn2 4260	. . 3 |- ( E! x E. y e. A x = B <-> E. z { x | E. y e. A x = B } = { z } )
2		eleq2 2690	. . . . . 6 |- ( { x | E. y e. A x = B } = { z } -> ( x e. { x | E. y e. A x = B } <-> x e. { z } ) )
3		abid 2610	. . . . . 6 |- ( x e. { x | E. y e. A x = B } <-> E. y e. A x = B )
4		velsn 4193	. . . . . 6 |- ( x e. { z } <-> x = z )
5	2, 3, 4	3bitr3g 302	. . . . 5 |- ( { x | E. y e. A x = B } = { z } -> ( E. y e. A x = B <-> x = z ) )
6		nfre1 3005	. . . . . . . . 9 |- F/ y E. y e. A x = B
7	6	nfab 2769	. . . . . . . 8 |- F/_ y { x | E. y e. A x = B }
8	7	nfeq1 2778	. . . . . . 7 |- F/ y { x | E. y e. A x = B } = { z }
9		eusvobj1.1	. . . . . . . . 9 |- B e. _V
10	9	elabrex 6501	. . . . . . . 8 |- ( y e. A -> B e. { x | E. y e. A x = B } )
11		eleq2 2690	. . . . . . . . 9 |- ( { x | E. y e. A x = B } = { z } -> ( B e. { x | E. y e. A x = B } <-> B e. { z } ) )
12	9	elsn 4192	. . . . . . . . . 10 |- ( B e. { z } <-> B = z )
13		eqcom 2629	. . . . . . . . . 10 |- ( B = z <-> z = B )
14	12, 13	bitri 264	. . . . . . . . 9 |- ( B e. { z } <-> z = B )
15	11, 14	syl6bb 276	. . . . . . . 8 |- ( { x | E. y e. A x = B } = { z } -> ( B e. { x | E. y e. A x = B } <-> z = B ) )
16	10, 15	syl5ib 234	. . . . . . 7 |- ( { x | E. y e. A x = B } = { z } -> ( y e. A -> z = B ) )
17	8, 16	ralrimi 2957	. . . . . 6 |- ( { x | E. y e. A x = B } = { z } -> A. y e. A z = B )
18		eqeq1 2626	. . . . . . 7 |- ( x = z -> ( x = B <-> z = B ) )
19	18	ralbidv 2986	. . . . . 6 |- ( x = z -> ( A. y e. A x = B <-> A. y e. A z = B ) )
20	17, 19	syl5ibrcom 237	. . . . 5 |- ( { x | E. y e. A x = B } = { z } -> ( x = z -> A. y e. A x = B ) )
21	5, 20	sylbid 230	. . . 4 |- ( { x | E. y e. A x = B } = { z } -> ( E. y e. A x = B -> A. y e. A x = B ) )
22	21	exlimiv 1858	. . 3 |- ( E. z { x | E. y e. A x = B } = { z } -> ( E. y e. A x = B -> A. y e. A x = B ) )
23	1, 22	sylbi 207	. 2 |- ( E! x E. y e. A x = B -> ( E. y e. A x = B -> A. y e. A x = B ) )
24		euex 2494	. . 3 |- ( E! x E. y e. A x = B -> E. x E. y e. A x = B )
25		rexn0 4074	. . . 4 |- ( E. y e. A x = B -> A =/= (/) )
26	25	exlimiv 1858	. . 3 |- ( E. x E. y e. A x = B -> A =/= (/) )
27		r19.2z 4060	. . . 4 |- ( ( A =/= (/) /\ A. y e. A x = B ) -> E. y e. A x = B )
28	27	ex 450	. . 3 |- ( A =/= (/) -> ( A. y e. A x = B -> E. y e. A x = B ) )
29	24, 26, 28	3syl 18	. 2 |- ( E! x E. y e. A x = B -> ( A. y e. A x = B -> E. y e. A x = B ) )
30	23, 29	impbid 202	1 |- ( E! x E. y e. A x = B -> ( E. y e. A x = B <-> A. y e. A x = B ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   E.wex 1704   e. wcel 1990   E!weu 2470   {cab 2608   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   (/)c0 3915   {csn 4177

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-nul 3916  df-sn 4178

This theorem is referenced by:  eusvobj1  6644