Theorem cnveq 4527

Description: Equality theorem for converse. (Contributed by NM, 13-Aug-1995.)

Assertion

Ref	Expression
cnveq	|- ( A = B -> `' A = `' B )

Proof of Theorem cnveq

Step	Hyp	Ref	Expression
1		cnvss 4526	. . 3 |- ( A C_ B -> `' A C_ `' B )
2		cnvss 4526	. . 3 |- ( B C_ A -> `' B C_ `' A )
3	1, 2	anim12i 331	. 2 |- ( ( A C_ B /\ B C_ A ) -> ( `' A C_ `' B /\ `' B C_ `' A ) )
4		eqss 3014	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
5		eqss 3014	. 2 |- ( `' A = `' B <-> ( `' A C_ `' B /\ `' B C_ `' A ) )
6	3, 4, 5	3imtr4i 199	1 |- ( A = B -> `' A = `' B )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   C_ wss 2973   `'ccnv 4362

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-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-in 2979  df-ss 2986  df-br 3786  df-opab 3840  df-cnv 4371

This theorem is referenced by:  cnveqi  4528  cnveqd  4529  rneq  4579  cnveqb  4796  funcnvuni  4988  f1eq1  5107  f1o00  5181  foeqcnvco  5450  tposfn2  5904  ereq1  6136  infeq3  6428