Theorem reseq12d 4820

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

Hypotheses

Ref	Expression
reseqd.1	⊢ (𝜑 → 𝐴 = 𝐵)
reseqd.2	⊢ (𝜑 → 𝐶 = 𝐷)

Assertion

Ref	Expression
reseq12d	⊢ (𝜑 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐷))

Proof of Theorem reseq12d

Step	Hyp	Ref	Expression
1		reseqd.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
2	1	reseq1d 4818	. 2 ⊢ (𝜑 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶))
3		reseqd.2	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
4	3	reseq2d 4819	. 2 ⊢ (𝜑 → (𝐵 ↾ 𝐶) = (𝐵 ↾ 𝐷))
5	2, 4	eqtrd 2172	1 ⊢ (𝜑 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐷))

Syntax hints:   → wi 4   = wceq 1331   ↾ cres 4541

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077  df-opab 3990  df-xp 4545  df-res 4551

This theorem is referenced by:  tfrlem3ag  6206  tfrlem3a  6207  tfrlemi1  6229  tfr1onlem3ag  6234  setsvalg  11989  isxms  12620  isms  12622