Theorem cnveq 4904

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 4903	. . 3 |- ( A C_ B -> `' A C_ `' B )
2		cnvss 4903	. . 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 3242	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
5		eqss 3242	. 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 1397   C_ wss 3200   `'ccnv 4724

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-in 3206  df-ss 3213  df-br 4089  df-opab 4151  df-cnv 4733

This theorem is referenced by:  cnveqi  4905  cnveqd  4906  rneq  4959  cnveqb  5192  funcnvuni  5399  f1eq1  5537  f1ssf1  5615  f1o00  5620  foeqcnvco  5930  tposfn2  6431  ereq1  6708  infeq3  7213  1arith  12939  isrim0  14174  psrbag  14682  psr1clfi  14701  iscn  14920  ishmeo  15027  istrl  16235