Theorem coeq12d 4768

Description: Equality deduction for composition of two classes. (Contributed by FL, 7-Jun-2012.)

Hypotheses

Ref	Expression
coeq12d.1	⊢ (𝜑 → 𝐴 = 𝐵)
coeq12d.2	⊢ (𝜑 → 𝐶 = 𝐷)

Assertion

Ref	Expression
coeq12d	⊢ (𝜑 → (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐷))

Proof of Theorem coeq12d

Step	Hyp	Ref	Expression
1		coeq12d.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
2	1	coeq1d 4765	. 2 ⊢ (𝜑 → (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶))
3		coeq12d.2	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
4	3	coeq2d 4766	. 2 ⊢ (𝜑 → (𝐵 ∘ 𝐶) = (𝐵 ∘ 𝐷))
5	2, 4	eqtrd 2198	1 ⊢ (𝜑 → (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐷))

Syntax hints:   → wi 4   = wceq 1343   ∘ ccom 4608

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-in 3122  df-ss 3129  df-br 3983  df-opab 4044  df-co 4613

This theorem is referenced by: (None)