Theorem reseq12i 5974

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

Hypotheses

Ref	Expression
reseqi.1	⊢ 𝐴 = 𝐵
reseqi.2	⊢ 𝐶 = 𝐷

Assertion

Ref	Expression
reseq12i	⊢ (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐷)

Proof of Theorem reseq12i

Step	Hyp	Ref	Expression
1		reseqi.1	. . 3 ⊢ 𝐴 = 𝐵
2	1	reseq1i 5972	. 2 ⊢ (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶)
3		reseqi.2	. . 3 ⊢ 𝐶 = 𝐷
4	3	reseq2i 5973	. 2 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ↾ 𝐷)
5	2, 4	eqtri 2785	1 ⊢ (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐷)

This proof depends on syntax axioms:   = wceq 1570   ↾ cres 5661

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-in 3909  df-opab 5172  df-xp 5665  df-res 5671

This theorem is used by:  cnvresid  6616  fprlem1  8303  dfoi  9487  frrlem15  9743  lubfval  18442  glbfval  18455  odulub  18499  oduglb  18501  dvlog  26896  dvlog2  26898  issubgr  29739  finsumvtxdg2size  30018  sitgclg  34861  fourierdlem57  46999  fourierdlem74  47016  fourierdlem75  47017