Theorem cnveq 4557

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 4556	. . 3 |- ( A C_ B -> `' A C_ `' B )
2		cnvss 4556	. . 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 3023	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
5		eqss 3023	. 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 1285   C_ wss 2982   `'ccnv 4390

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-in 2988  df-ss 2995  df-br 3806  df-opab 3860  df-cnv 4399

This theorem is referenced by:  cnveqi  4558  cnveqd  4559  rneq  4609  cnveqb  4826  funcnvuni  5019  f1eq1  5138  f1o00  5212  foeqcnvco  5481  tposfn2  5935  ereq1  6200  infeq3  6522