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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 5723 | . 2 ⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶) |
3 | coeq12i.2 | . . 3 ⊢ 𝐶 = 𝐷 | |
4 | 3 | coeq2i 5724 | . 2 ⊢ (𝐵 ∘ 𝐶) = (𝐵 ∘ 𝐷) |
5 | 2, 4 | eqtri 2841 | 1 ⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∘ ccom 5552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-in 3940 df-ss 3949 df-br 5058 df-opab 5120 df-co 5557 |
This theorem is referenced by: madetsumid 20998 mdetleib2 21125 imsval 28389 pjcmul1i 29905 coprprop 30361 cotrcltrcl 39948 brtrclfv2 39950 clsneif1o 40332 |
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