Theorem cnveqb 5590

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 5296	. 2 |- ( A = B -> `' A = `' B )
2		dfrel2 5583	. . . 4 |- ( Rel A <-> `' `' A = A )
3		dfrel2 5583	. . . . . . 7 |- ( Rel B <-> `' `' B = B )
4		cnveq 5296	. . . . . . . . 9 |- ( `' A = `' B -> `' `' A = `' `' B )
5		eqeq2 2633	. . . . . . . . 9 |- ( B = `' `' B -> ( `' `' A = B <-> `' `' A = `' `' B ) )
6	4, 5	syl5ibr 236	. . . . . . . 8 |- ( B = `' `' B -> ( `' A = `' B -> `' `' A = B ) )
7	6	eqcoms 2630	. . . . . . 7 |- ( `' `' B = B -> ( `' A = `' B -> `' `' A = B ) )
8	3, 7	sylbi 207	. . . . . 6 |- ( Rel B -> ( `' A = `' B -> `' `' A = B ) )
9		eqeq1 2626	. . . . . . 7 |- ( A = `' `' A -> ( A = B <-> `' `' A = B ) )
10	9	imbi2d 330	. . . . . 6 |- ( A = `' `' A -> ( ( `' A = `' B -> A = B ) <-> ( `' A = `' B -> `' `' A = B ) ) )
11	8, 10	syl5ibr 236	. . . . 5 |- ( A = `' `' A -> ( Rel B -> ( `' A = `' B -> A = B ) ) )
12	11	eqcoms 2630	. . . 4 |- ( `' `' A = A -> ( Rel B -> ( `' A = `' B -> A = B ) ) )
13	2, 12	sylbi 207	. . 3 |- ( Rel A -> ( Rel B -> ( `' A = `' B -> A = B ) ) )
14	13	imp 445	. 2 |- ( ( Rel A /\ Rel B ) -> ( `' A = `' B -> A = B ) )
15	1, 14	impbid2 216	1 |- ( ( Rel A /\ Rel B ) -> ( A = B <-> `' A = `' B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   `'ccnv 5113   Rel wrel 5119

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  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  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-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122

This theorem is referenced by:  cnveq0  5591  weisoeq2  6606  relexpaddg  13793  relexpaddss  38010