Theorem cnveq 4721

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 4720	. . 3 |- ( A C_ B -> `' A C_ `' B )
2		cnvss 4720	. . 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 3117	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
5		eqss 3117	. 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 1332   C_ wss 3076   `'ccnv 4546

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-in 3082  df-ss 3089  df-br 3938  df-opab 3998  df-cnv 4555

This theorem is referenced by:  cnveqi  4722  cnveqd  4723  rneq  4774  cnveqb  5002  funcnvuni  5200  f1eq1  5331  f1o00  5410  foeqcnvco  5699  tposfn2  6171  ereq1  6444  infeq3  6910  iscn  12405  ishmeo  12512