Theorem reseq2d 4630

Description: Equality deduction for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)

Hypothesis

Ref	Expression
reseqd.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
reseq2d	⊢ (𝜑 → (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵))

Proof of Theorem reseq2d

Step	Hyp	Ref	Expression
1		reseqd.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		reseq2 4625	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵))
3	1, 2	syl 14	1 ⊢ (𝜑 → (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵))

Syntax hints:   → wi 4   = wceq 1284   ↾ cres 4365

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-in 2979  df-opab 3840  df-xp 4369  df-res 4375

This theorem is referenced by:  reseq12d  4631  resima2  4662  relresfld  4867  f1orescnv  5162  funcocnv2  5171  fococnv2  5172  fnressn  5370  oprssov  5662  dftpos2  5899  dif1en  6364  fseq1p1m1  9111