Theorem coideq 38569

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 5812	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∘ 𝐴) = (𝐵 ∘ 𝐴))
2		coeq2 5813	. 2 ⊢ (𝐴 = 𝐵 → (𝐵 ∘ 𝐴) = (𝐵 ∘ 𝐵))
3	1, 2	eqtrd 2771	1 ⊢ (𝐴 = 𝐵 → (𝐴 ∘ 𝐴) = (𝐵 ∘ 𝐵))

Syntax hints:   → wi 4   = wceq 1542   ∘ ccom 5635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ss 3906  df-br 5086  df-opab 5148  df-co 5640

This theorem is referenced by:  eltrrels2  38984  trreleq  38987  eleqvrels2  38997