Theorem cnveqb 4796

Description: Equality theorem for converse. (Contributed by FL, 19-Sep-2011.)

Assertion

Ref	Expression
cnveqb	|- ( ( Rel A /\ Rel B ) -> ( A = B <-> `' A = `' B ) )

Proof of Theorem cnveqb

Step	Hyp	Ref	Expression
1		cnveq 4527	. 2 |- ( A = B -> `' A = `' B )
2		dfrel2 4791	. . . 4 |- ( Rel A <-> `' `' A = A )
3		dfrel2 4791	. . . . . . 7 |- ( Rel B <-> `' `' B = B )
4		cnveq 4527	. . . . . . . . 9 |- ( `' A = `' B -> `' `' A = `' `' B )
5		eqeq2 2090	. . . . . . . . 9 |- ( B = `' `' B -> ( `' `' A = B <-> `' `' A = `' `' B ) )
6	4, 5	syl5ibr 154	. . . . . . . 8 |- ( B = `' `' B -> ( `' A = `' B -> `' `' A = B ) )
7	6	eqcoms 2084	. . . . . . 7 |- ( `' `' B = B -> ( `' A = `' B -> `' `' A = B ) )
8	3, 7	sylbi 119	. . . . . 6 |- ( Rel B -> ( `' A = `' B -> `' `' A = B ) )
9		eqeq1 2087	. . . . . . 7 |- ( A = `' `' A -> ( A = B <-> `' `' A = B ) )
10	9	imbi2d 228	. . . . . 6 |- ( A = `' `' A -> ( ( `' A = `' B -> A = B ) <-> ( `' A = `' B -> `' `' A = B ) ) )
11	8, 10	syl5ibr 154	. . . . 5 |- ( A = `' `' A -> ( Rel B -> ( `' A = `' B -> A = B ) ) )
12	11	eqcoms 2084	. . . 4 |- ( `' `' A = A -> ( Rel B -> ( `' A = `' B -> A = B ) ) )
13	2, 12	sylbi 119	. . 3 |- ( Rel A -> ( Rel B -> ( `' A = `' B -> A = B ) ) )
14	13	imp 122	. 2 |- ( ( Rel A /\ Rel B ) -> ( `' A = `' B -> A = B ) )
15	1, 14	impbid2 141	1 |- ( ( Rel A /\ Rel B ) -> ( A = B <-> `' A = `' B ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   `'ccnv 4362   Rel wrel 4368

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-rel 4370  df-cnv 4371

This theorem is referenced by:  cnveq0  4797