Theorem cnveq 4853

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 4852	. . 3 |- ( A C_ B -> `' A C_ `' B )
2		cnvss 4852	. . 3 |- ( B C_ A -> `' B C_ `' A )
3	1, 2	anim12i 338	. 2 |- ( ( A C_ B /\ B C_ A ) -> ( `' A C_ `' B /\ `' B C_ `' A ) )
4		eqss 3208	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
5		eqss 3208	. 2 |- ( `' A = `' B <-> ( `' A C_ `' B /\ `' B C_ `' A ) )
6	3, 4, 5	3imtr4i 201	1 |- ( A = B -> `' A = `' B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   C_ wss 3166   `'ccnv 4675

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-in 3172  df-ss 3179  df-br 4046  df-opab 4107  df-cnv 4684

This theorem is referenced by:  cnveqi  4854  cnveqd  4855  rneq  4906  cnveqb  5139  funcnvuni  5344  f1eq1  5478  f1o00  5559  foeqcnvco  5861  tposfn2  6354  ereq1  6629  infeq3  7119  1arith  12723  isrim0  13956  psrbag  14464  psr1clfi  14483  iscn  14702  ishmeo  14809