Theorem coideq 34510

Description: Equality theorem for composition of two classes. (Contributed by Peter Mazsa, 23-Sep-2021.)

Assertion

Ref	Expression
coideq	⊢ (𝐴 = 𝐵 → (𝐴 ∘ 𝐴) = (𝐵 ∘ 𝐵))

Proof of Theorem coideq

Step	Hyp	Ref	Expression
1		coeq1 5483	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∘ 𝐴) = (𝐵 ∘ 𝐴))
2		coeq2 5484	. 2 ⊢ (𝐴 = 𝐵 → (𝐵 ∘ 𝐴) = (𝐵 ∘ 𝐵))
3	1, 2	eqtrd 2833	1 ⊢ (𝐴 = 𝐵 → (𝐴 ∘ 𝐴) = (𝐵 ∘ 𝐵))

Syntax hints:   → wi 4   = wceq 1653   ∘ ccom 5316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-in 3776  df-ss 3783  df-br 4844  df-opab 4906  df-co 5321

This theorem is referenced by:  eltrrels2  34819  trreleq  34822  eleqvrels2  34830