Theorem cnveqd 4785

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 4783	. 2 ⊢ (𝐴 = 𝐵 → ◡𝐴 = ◡𝐵)
3	1, 2	syl 14	1 ⊢ (𝜑 → ◡𝐴 = ◡𝐵)

Syntax hints:   → wi 4   = wceq 1348  ^◡ccnv 4608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-in 3127  df-ss 3134  df-br 3988  df-opab 4049  df-cnv 4617

This theorem is referenced by:  cnvsng  5094  cores2  5121  suppssof1  6076  2ndval2  6133  2nd1st  6157  cnvf1olem  6201  brtpos2  6228  dftpos4  6240  tpostpos  6241  tposf12  6246  xpcomco  6802  infeq123d  6991  fsumcnv  11393  fprodcnv  11581  ennnfonelemf1  12366  txswaphmeolem  13079