Theorem cnveq 4778

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 4777	. . 3 |- ( A C_ B -> `' A C_ `' B )
2		cnvss 4777	. . 3 |- ( B C_ A -> `' B C_ `' A )
3	1, 2	anim12i 336	. 2 |- ( ( A C_ B /\ B C_ A ) -> ( `' A C_ `' B /\ `' B C_ `' A ) )
4		eqss 3157	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
5		eqss 3157	. 2 |- ( `' A = `' B <-> ( `' A C_ `' B /\ `' B C_ `' A ) )
6	3, 4, 5	3imtr4i 200	1 |- ( A = B -> `' A = `' B )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   C_ wss 3116   `'ccnv 4603

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-in 3122  df-ss 3129  df-br 3983  df-opab 4044  df-cnv 4612

This theorem is referenced by:  cnveqi  4779  cnveqd  4780  rneq  4831  cnveqb  5059  funcnvuni  5257  f1eq1  5388  f1o00  5467  foeqcnvco  5758  tposfn2  6234  ereq1  6508  infeq3  6980  1arith  12297  iscn  12837  ishmeo  12944