Theorem eusvobj2 5714

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 3558	. . 3 |- ( E! x E. y e. A x = B <-> E. z { x | E. y e. A x = B } = { z } )
2		eleq2 2178	. . . . . 6 |- ( { x | E. y e. A x = B } = { z } -> ( x e. { x | E. y e. A x = B } <-> x e. { z } ) )
3		abid 2103	. . . . . 6 |- ( x e. { x | E. y e. A x = B } <-> E. y e. A x = B )
4		velsn 3510	. . . . . 6 |- ( x e. { z } <-> x = z )
5	2, 3, 4	3bitr3g 221	. . . . 5 |- ( { x | E. y e. A x = B } = { z } -> ( E. y e. A x = B <-> x = z ) )
6		nfre1 2450	. . . . . . . . 9 |- F/ y E. y e. A x = B
7	6	nfab 2260	. . . . . . . 8 |- F/_ y { x | E. y e. A x = B }
8	7	nfeq1 2265	. . . . . . 7 |- F/ y { x | E. y e. A x = B } = { z }
9		eusvobj1.1	. . . . . . . . 9 |- B e. _V
10	9	elabrex 5613	. . . . . . . 8 |- ( y e. A -> B e. { x | E. y e. A x = B } )
11		eleq2 2178	. . . . . . . . 9 |- ( { x | E. y e. A x = B } = { z } -> ( B e. { x | E. y e. A x = B } <-> B e. { z } ) )
12	9	elsn 3509	. . . . . . . . . 10 |- ( B e. { z } <-> B = z )
13		eqcom 2117	. . . . . . . . . 10 |- ( B = z <-> z = B )
14	12, 13	bitri 183	. . . . . . . . 9 |- ( B e. { z } <-> z = B )
15	11, 14	syl6bb 195	. . . . . . . 8 |- ( { x | E. y e. A x = B } = { z } -> ( B e. { x | E. y e. A x = B } <-> z = B ) )
16	10, 15	syl5ib 153	. . . . . . 7 |- ( { x | E. y e. A x = B } = { z } -> ( y e. A -> z = B ) )
17	8, 16	ralrimi 2477	. . . . . 6 |- ( { x | E. y e. A x = B } = { z } -> A. y e. A z = B )
18		eqeq1 2121	. . . . . . 7 |- ( x = z -> ( x = B <-> z = B ) )
19	18	ralbidv 2411	. . . . . 6 |- ( x = z -> ( A. y e. A x = B <-> A. y e. A z = B ) )
20	17, 19	syl5ibrcom 156	. . . . 5 |- ( { x | E. y e. A x = B } = { z } -> ( x = z -> A. y e. A x = B ) )
21	5, 20	sylbid 149	. . . 4 |- ( { x | E. y e. A x = B } = { z } -> ( E. y e. A x = B -> A. y e. A x = B ) )
22	21	exlimiv 1560	. . 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 120	. 2 |- ( E! x E. y e. A x = B -> ( E. y e. A x = B -> A. y e. A x = B ) )
24		euex 2005	. . 3 |- ( E! x E. y e. A x = B -> E. x E. y e. A x = B )
25		rexm 3428	. . . 4 |- ( E. y e. A x = B -> E. y y e. A )
26	25	exlimiv 1560	. . 3 |- ( E. x E. y e. A x = B -> E. y y e. A )
27		r19.2m 3415	. . . 4 |- ( ( E. y y e. A /\ A. y e. A x = B ) -> E. y e. A x = B )
28	27	ex 114	. . 3 |- ( E. y y e. A -> ( A. y e. A x = B -> E. y e. A x = B ) )
29	24, 26, 28	3syl 17	. 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 128	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 104   = wceq 1314   E.wex 1451   e. wcel 1463   E!weu 1975   {cab 2101   A.wral 2390   E.wrex 2391   _Vcvv 2657   {csn 3493

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-sbc 2879  df-csb 2972  df-sn 3499

This theorem is referenced by:  eusvobj1  5715