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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 5518 | . 2 ⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶) |
3 | coeq12i.2 | . . 3 ⊢ 𝐶 = 𝐷 | |
4 | 3 | coeq2i 5519 | . 2 ⊢ (𝐵 ∘ 𝐶) = (𝐵 ∘ 𝐷) |
5 | 2, 4 | eqtri 2849 | 1 ⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1656 ∘ ccom 5350 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-in 3805 df-ss 3812 df-br 4876 df-opab 4938 df-co 5355 |
This theorem is referenced by: madetsumid 20642 mdetleib2 20769 imsval 28091 pjcmul1i 29611 cotrcltrcl 38853 brtrclfv2 38855 clsneif1o 39237 |
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