Theorem cnveqd 4454

Description: Equality deduction for converse. (Contributed by NM, 6-Dec-2013.)

Hypothesis

Ref	Expression
cnveqd.1	⊢ (φ → A = B)

Assertion

Ref	Expression
cnveqd	⊢ (φ → ◡A = ◡B)

Proof of Theorem cnveqd

Step	Hyp	Ref	Expression
1		cnveqd.1	. 2 ⊢ (φ → A = B)
2		cnveq 4452	. 2 ⊢ (A = B → ◡A = ◡B)
3	1, 2	syl 14	1 ⊢ (φ → ◡A = ◡B)

Syntax hints:   → wi 4   = wceq 1242  ^◡ccnv 4287

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-in 2918  df-ss 2925  df-br 3756  df-opab 3810  df-cnv 4296

This theorem is referenced by:  cnvsng  4749  cores2  4776  suppssof1  5670  2ndval2  5725  2nd1st  5748  cnvf1olem  5787  brtpos2  5807  dftpos4  5819  tpostpos  5820  tposf12  5825  xpcomco  6236