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Mirrors > Home > MPE Home > Th. List > Mathboxes > coideq | Structured version Visualization version GIF version |
Description: Equality theorem for composition of two classes. (Contributed by Peter Mazsa, 23-Sep-2021.) |
Ref | Expression |
---|---|
coideq | ⊢ (𝐴 = 𝐵 → (𝐴 ∘ 𝐴) = (𝐵 ∘ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coeq1 5723 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∘ 𝐴) = (𝐵 ∘ 𝐴)) | |
2 | coeq2 5724 | . 2 ⊢ (𝐴 = 𝐵 → (𝐵 ∘ 𝐴) = (𝐵 ∘ 𝐵)) | |
3 | 1, 2 | eqtrd 2856 | 1 ⊢ (𝐴 = 𝐵 → (𝐴 ∘ 𝐴) = (𝐵 ∘ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∘ ccom 5554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-in 3943 df-ss 3952 df-br 5060 df-opab 5122 df-co 5559 |
This theorem is referenced by: eltrrels2 35809 trreleq 35812 eleqvrels2 35821 |
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