Theorem cnveq 4887

Description: Equality theorem for converse. (Contributed by set.mm contributors, 13-Aug-1995.)

Assertion

Ref	Expression
cnveq	|- (A = B -> `'A = `'B)

Proof of Theorem cnveq

Step	Hyp	Ref	Expression
1		cnvss 4886	. . 3 |- (A C_ B -> `'A C_ `'B)
2		cnvss 4886	. . 3 |- (B C_ A -> `'B C_ `'A)
3	1, 2	anim12i 549	. 2 |- ((A C_ B /\ B C_ A) -> (`'A C_ `'B /\ `'B C_ `'A))
4		eqss 3288	. 2 |- (A = B <-> (A C_ B /\ B C_ A))
5		eqss 3288	. 2 |- (`'A = `'B <-> (`'A C_ `'B /\ `'B C_ `'A))
6	3, 4, 5	3imtr4i 257	1 |- (A = B -> `'A = `'B)

Syntax hints:   -> wi 4   /\ wa 358   = wceq 1642   C_ wss 3258  `'ccnv 4772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-opab 4624  df-br 4641  df-cnv 4786

This theorem is referenced by:  cnveqi  4888  cnveqd  4889  cnveqb  5064  funcnvuni  5162  f1eq1  5254  f1o00  5318  enprmaplem3  6079  enprmaplem5  6081  enprmaplem6  6082