Theorem cnveqb 5066

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 4785	. 2 |- ( A = B -> `' A = `' B )
2		dfrel2 5061	. . . 4 |- ( Rel A <-> `' `' A = A )
3		dfrel2 5061	. . . . . . 7 |- ( Rel B <-> `' `' B = B )
4		cnveq 4785	. . . . . . . . 9 |- ( `' A = `' B -> `' `' A = `' `' B )
5		eqeq2 2180	. . . . . . . . 9 |- ( B = `' `' B -> ( `' `' A = B <-> `' `' A = `' `' B ) )
6	4, 5	syl5ibr 155	. . . . . . . 8 |- ( B = `' `' B -> ( `' A = `' B -> `' `' A = B ) )
7	6	eqcoms 2173	. . . . . . 7 |- ( `' `' B = B -> ( `' A = `' B -> `' `' A = B ) )
8	3, 7	sylbi 120	. . . . . 6 |- ( Rel B -> ( `' A = `' B -> `' `' A = B ) )
9		eqeq1 2177	. . . . . . 7 |- ( A = `' `' A -> ( A = B <-> `' `' A = B ) )
10	9	imbi2d 229	. . . . . 6 |- ( A = `' `' A -> ( ( `' A = `' B -> A = B ) <-> ( `' A = `' B -> `' `' A = B ) ) )
11	8, 10	syl5ibr 155	. . . . 5 |- ( A = `' `' A -> ( Rel B -> ( `' A = `' B -> A = B ) ) )
12	11	eqcoms 2173	. . . 4 |- ( `' `' A = A -> ( Rel B -> ( `' A = `' B -> A = B ) ) )
13	2, 12	sylbi 120	. . 3 |- ( Rel A -> ( Rel B -> ( `' A = `' B -> A = B ) ) )
14	13	imp 123	. 2 |- ( ( Rel A /\ Rel B ) -> ( `' A = `' B -> A = B ) )
15	1, 14	impbid2 142	1 |- ( ( Rel A /\ Rel B ) -> ( A = B <-> `' A = `' B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   `'ccnv 4610   Rel wrel 4616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619

This theorem is referenced by:  cnveq0  5067