Theorem cnveq 4785

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 4784	. . 3 |- ( A C_ B -> `' A C_ `' B )
2		cnvss 4784	. . 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 3162	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
5		eqss 3162	. 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 1348   C_ wss 3121   `'ccnv 4610

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-in 3127  df-ss 3134  df-br 3990  df-opab 4051  df-cnv 4619

This theorem is referenced by:  cnveqi  4786  cnveqd  4787  rneq  4838  cnveqb  5066  funcnvuni  5267  f1eq1  5398  f1o00  5477  foeqcnvco  5769  tposfn2  6245  ereq1  6520  infeq3  6992  1arith  12319  iscn  12991  ishmeo  13098