Theorem cnveqi 4841

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 4840	. 2 ⊢ (𝐴 = 𝐵 → ◡𝐴 = ◡𝐵)
3	1, 2	ax-mp 5	1 ⊢ ◡𝐴 = ◡𝐵

Syntax hints:   = wceq 1364  ^◡ccnv 4662

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-in 3163  df-ss 3170  df-br 4034  df-opab 4095  df-cnv 4671

This theorem is referenced by:  mptcnv  5072  cnvxp  5088  xp0  5089  imainrect  5115  cnvcnv  5122  mptpreima  5163  co01  5184  coi2  5186  cocnvres  5194  fcoi1  5438  fun11iun  5525  f1ocnvd  6125  cnvoprab  6292  f1od2  6293  mapsncnv  6754  sbthlemi8  7030  caseinj  7155  djuinj  7172  fisumcom2  11603  fprodcom2fi  11791