Theorem cnveq 4837

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 4836	. . 3 |- ( A C_ B -> `' A C_ `' B )
2		cnvss 4836	. . 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 3195	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
5		eqss 3195	. 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 3154   `'ccnv 4659

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-in 3160  df-ss 3167  df-br 4031  df-opab 4092  df-cnv 4668

This theorem is referenced by:  cnveqi  4838  cnveqd  4839  rneq  4890  cnveqb  5122  funcnvuni  5324  f1eq1  5455  f1o00  5536  foeqcnvco  5834  tposfn2  6321  ereq1  6596  infeq3  7076  1arith  12508  isrim0  13660  psrbag  14166  iscn  14376  ishmeo  14483