Theorem cnveq 4803

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 4802	. . 3 |- ( A C_ B -> `' A C_ `' B )
2		cnvss 4802	. . 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 3172	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
5		eqss 3172	. 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 1353   C_ wss 3131   `'ccnv 4627

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-in 3137  df-ss 3144  df-br 4006  df-opab 4067  df-cnv 4636

This theorem is referenced by:  cnveqi  4804  cnveqd  4805  rneq  4856  cnveqb  5086  funcnvuni  5287  f1eq1  5418  f1o00  5498  foeqcnvco  5793  tposfn2  6269  ereq1  6544  infeq3  7016  1arith  12367  iscn  13736  ishmeo  13843