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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 5724 | . 2 ⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶) |
3 | coeq12i.2 | . . 3 ⊢ 𝐶 = 𝐷 | |
4 | 3 | coeq2i 5725 | . 2 ⊢ (𝐵 ∘ 𝐶) = (𝐵 ∘ 𝐷) |
5 | 2, 4 | eqtri 2844 | 1 ⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∘ ccom 5553 |
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 2157 ax-12 2173 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 3942 df-ss 3951 df-br 5059 df-opab 5121 df-co 5558 |
This theorem is referenced by: madetsumid 21064 mdetleib2 21191 imsval 28456 pjcmul1i 29972 coprprop 30429 cotrcltrcl 40063 brtrclfv2 40065 clsneif1o 40447 |
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