Theorem cnveqb 5083

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 4800	. 2 |- ( A = B -> `' A = `' B )
2		dfrel2 5078	. . . 4 |- ( Rel A <-> `' `' A = A )
3		dfrel2 5078	. . . . . . 7 |- ( Rel B <-> `' `' B = B )
4		cnveq 4800	. . . . . . . . 9 |- ( `' A = `' B -> `' `' A = `' `' B )
5		eqeq2 2187	. . . . . . . . 9 |- ( B = `' `' B -> ( `' `' A = B <-> `' `' A = `' `' B ) )
6	4, 5	imbitrrid 156	. . . . . . . 8 |- ( B = `' `' B -> ( `' A = `' B -> `' `' A = B ) )
7	6	eqcoms 2180	. . . . . . 7 |- ( `' `' B = B -> ( `' A = `' B -> `' `' A = B ) )
8	3, 7	sylbi 121	. . . . . 6 |- ( Rel B -> ( `' A = `' B -> `' `' A = B ) )
9		eqeq1 2184	. . . . . . 7 |- ( A = `' `' A -> ( A = B <-> `' `' A = B ) )
10	9	imbi2d 230	. . . . . 6 |- ( A = `' `' A -> ( ( `' A = `' B -> A = B ) <-> ( `' A = `' B -> `' `' A = B ) ) )
11	8, 10	imbitrrid 156	. . . . 5 |- ( A = `' `' A -> ( Rel B -> ( `' A = `' B -> A = B ) ) )
12	11	eqcoms 2180	. . . 4 |- ( `' `' A = A -> ( Rel B -> ( `' A = `' B -> A = B ) ) )
13	2, 12	sylbi 121	. . 3 |- ( Rel A -> ( Rel B -> ( `' A = `' B -> A = B ) ) )
14	13	imp 124	. 2 |- ( ( Rel A /\ Rel B ) -> ( `' A = `' B -> A = B ) )
15	1, 14	impbid2 143	1 |- ( ( Rel A /\ Rel B ) -> ( A = B <-> `' A = `' B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   `'ccnv 4624   Rel wrel 4630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4003  df-opab 4064  df-xp 4631  df-rel 4632  df-cnv 4633

This theorem is referenced by:  cnveq0  5084