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Mirrors > Home > MPE Home > Th. List > coeq12i | Structured version Visualization version GIF version |
Description: Equality inference for composition of two classes. (Contributed by FL, 7-Jun-2012.) |
Ref | Expression |
---|---|
coeq12i.1 | ⊢ 𝐴 = 𝐵 |
coeq12i.2 | ⊢ 𝐶 = 𝐷 |
Ref | Expression |
---|---|
coeq12i | ⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coeq12i.1 | . . 3 ⊢ 𝐴 = 𝐵 | |
2 | 1 | coeq1i 5816 | . 2 ⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶) |
3 | coeq12i.2 | . . 3 ⊢ 𝐶 = 𝐷 | |
4 | 3 | coeq2i 5817 | . 2 ⊢ (𝐵 ∘ 𝐶) = (𝐵 ∘ 𝐷) |
5 | 2, 4 | eqtri 2761 | 1 ⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∘ ccom 5638 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3446 df-in 3918 df-ss 3928 df-br 5107 df-opab 5169 df-co 5643 |
This theorem is referenced by: madetsumid 21826 mdetleib2 21953 imsval 29669 pjcmul1i 31185 coprprop 31660 cotrcltrcl 42085 brtrclfv2 42087 clsneif1o 42464 |
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