Theorem cnveqi 4872

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 4871	. 2 ⊢ (𝐴 = 𝐵 → ◡𝐴 = ◡𝐵)
3	1, 2	ax-mp 5	1 ⊢ ◡𝐴 = ◡𝐵

Syntax hints:   = wceq 1373  ^◡ccnv 4693

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-in 3181  df-ss 3188  df-br 4061  df-opab 4123  df-cnv 4702

This theorem is referenced by:  mptcnv  5105  cnvxp  5121  xp0  5122  imainrect  5148  cnvcnv  5155  mptpreima  5196  co01  5217  coi2  5219  cocnvres  5227  fcoi1  5479  fun11iun  5566  f1ocnvd  6173  cnvoprab  6345  f1od2  6346  mapsncnv  6807  sbthlemi8  7094  caseinj  7219  djuinj  7236  fisumcom2  11910  fprodcom2fi  12098