Theorem cnveqi 4800

Description: Equality inference for converse. (Contributed by NM, 23-Dec-2008.)

Hypothesis

Ref	Expression
cnveqi.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
cnveqi	⊢ ◡𝐴 = ◡𝐵

Proof of Theorem cnveqi

Step	Hyp	Ref	Expression
1		cnveqi.1	. 2 ⊢ 𝐴 = 𝐵
2		cnveq 4799	. 2 ⊢ (𝐴 = 𝐵 → ◡𝐴 = ◡𝐵)
3	1, 2	ax-mp 5	1 ⊢ ◡𝐴 = ◡𝐵

Syntax hints:   = wceq 1353  ^◡ccnv 4624

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 3135  df-ss 3142  df-br 4003  df-opab 4064  df-cnv 4633

This theorem is referenced by:  mptcnv  5029  cnvxp  5045  xp0  5046  imainrect  5072  cnvcnv  5079  mptpreima  5120  co01  5141  coi2  5143  cocnvres  5151  fcoi1  5394  fun11iun  5480  f1ocnvd  6069  cnvoprab  6231  f1od2  6232  mapsncnv  6691  sbthlemi8  6959  caseinj  7084  djuinj  7101  fisumcom2  11438  fprodcom2fi  11626