Theorem reseq12d 4936

Description: Equality deduction for restrictions. (Contributed by set.mm contributors, 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 4934	. 2 ⊢ (φ → (A ↾ C) = (B ↾ C))
3		reseqd.2	. . 3 ⊢ (φ → C = D)
4	3	reseq2d 4935	. 2 ⊢ (φ → (B ↾ C) = (B ↾ D))
5	2, 4	eqtrd 2385	1 ⊢ (φ → (A ↾ C) = (B ↾ D))

Syntax hints:   → wi 4   = wceq 1642   ↾ cres 4775

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-opab 4624  df-xp 4785  df-res 4789

This theorem is referenced by: (None)