Theorem reseq12d 4710

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 4708	. 2 ⊢ (𝜑 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶))
3		reseqd.2	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
4	3	reseq2d 4709	. 2 ⊢ (𝜑 → (𝐵 ↾ 𝐶) = (𝐵 ↾ 𝐷))
5	2, 4	eqtrd 2120	1 ⊢ (𝜑 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐷))

Syntax hints:   → wi 4   = wceq 1289   ↾ cres 4438

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3005  df-opab 3898  df-xp 4442  df-res 4448

This theorem is referenced by:  tfrlem3ag  6066  tfrlem3a  6067  tfrlemi1  6089  tfr1onlem3ag  6094