Theorem cnveq 4841

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 4840	. . 3 |- ( A C_ B -> `' A C_ `' B )
2		cnvss 4840	. . 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 3199	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
5		eqss 3199	. 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 1364   C_ wss 3157   `'ccnv 4663

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-in 3163  df-ss 3170  df-br 4035  df-opab 4096  df-cnv 4672

This theorem is referenced by:  cnveqi  4842  cnveqd  4843  rneq  4894  cnveqb  5126  funcnvuni  5328  f1eq1  5461  f1o00  5542  foeqcnvco  5840  tposfn2  6333  ereq1  6608  infeq3  7090  1arith  12561  isrim0  13793  psrbag  14299  psr1clfi  14316  iscn  14517  ishmeo  14624