Theorem reseq1d 4611

Description: Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)

Hypothesis

Ref	Expression
reseqd.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
reseq1d	⊢ (𝜑 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶))

Proof of Theorem reseq1d

Step	Hyp	Ref	Expression
1		reseqd.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		reseq1 4606	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶))
3	1, 2	syl 14	1 ⊢ (𝜑 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶))

Syntax hints:   → wi 4   = wceq 1243   ↾ cres 4347

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-res 4357

This theorem is referenced by:  reseq12d  4613  fun2ssres  4943  funcnvres2  4974  funimaexg  4983  fresin  5068  offres  5762  tfrlemisucaccv  5939  tfrlemi1  5946  freceq1  5979  freceq2  5980  fseq1p1m1  8956