Theorem cnveqb 5121

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 4836	. 2 |- ( A = B -> `' A = `' B )
2		dfrel2 5116	. . . 4 |- ( Rel A <-> `' `' A = A )
3		dfrel2 5116	. . . . . . 7 |- ( Rel B <-> `' `' B = B )
4		cnveq 4836	. . . . . . . . 9 |- ( `' A = `' B -> `' `' A = `' `' B )
5		eqeq2 2203	. . . . . . . . 9 |- ( B = `' `' B -> ( `' `' A = B <-> `' `' A = `' `' B ) )
6	4, 5	imbitrrid 156	. . . . . . . 8 |- ( B = `' `' B -> ( `' A = `' B -> `' `' A = B ) )
7	6	eqcoms 2196	. . . . . . 7 |- ( `' `' B = B -> ( `' A = `' B -> `' `' A = B ) )
8	3, 7	sylbi 121	. . . . . 6 |- ( Rel B -> ( `' A = `' B -> `' `' A = B ) )
9		eqeq1 2200	. . . . . . 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 2196	. . . 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 1364   `'ccnv 4658   Rel wrel 4664

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4665  df-rel 4666  df-cnv 4667

This theorem is referenced by:  cnveq0  5122