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Mirrors > Home > MPE Home > Th. List > Mathboxes > csbcog | Structured version Visualization version GIF version |
Description: Distribute proper substitution through a composition of relations. (Contributed by RP, 28-Jun-2020.) |
Ref | Expression |
---|---|
csbcog | ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵 ∘ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∘ ⦋𝐴 / 𝑥⦌𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbeq1 3886 | . . 3 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌(𝐵 ∘ 𝐶) = ⦋𝐴 / 𝑥⦌(𝐵 ∘ 𝐶)) | |
2 | csbeq1 3886 | . . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵) | |
3 | csbeq1 3886 | . . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐶) | |
4 | 2, 3 | coeq12d 5730 | . . 3 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌𝐵 ∘ ⦋𝑦 / 𝑥⦌𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∘ ⦋𝐴 / 𝑥⦌𝐶)) |
5 | 1, 4 | eqeq12d 2837 | . 2 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌(𝐵 ∘ 𝐶) = (⦋𝑦 / 𝑥⦌𝐵 ∘ ⦋𝑦 / 𝑥⦌𝐶) ↔ ⦋𝐴 / 𝑥⦌(𝐵 ∘ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∘ ⦋𝐴 / 𝑥⦌𝐶))) |
6 | vex 3498 | . . 3 ⊢ 𝑦 ∈ V | |
7 | nfcsb1v 3907 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | |
8 | nfcsb1v 3907 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶 | |
9 | 7, 8 | nfco 5731 | . . 3 ⊢ Ⅎ𝑥(⦋𝑦 / 𝑥⦌𝐵 ∘ ⦋𝑦 / 𝑥⦌𝐶) |
10 | csbeq1a 3897 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
11 | csbeq1a 3897 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶) | |
12 | 10, 11 | coeq12d 5730 | . . 3 ⊢ (𝑥 = 𝑦 → (𝐵 ∘ 𝐶) = (⦋𝑦 / 𝑥⦌𝐵 ∘ ⦋𝑦 / 𝑥⦌𝐶)) |
13 | 6, 9, 12 | csbief 3917 | . 2 ⊢ ⦋𝑦 / 𝑥⦌(𝐵 ∘ 𝐶) = (⦋𝑦 / 𝑥⦌𝐵 ∘ ⦋𝑦 / 𝑥⦌𝐶) |
14 | 5, 13 | vtoclg 3568 | 1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵 ∘ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∘ ⦋𝐴 / 𝑥⦌𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ⦋csb 3883 ∘ 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-3an 1085 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-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-br 5060 df-opab 5122 df-co 5559 |
This theorem is referenced by: brtrclfv2 40065 |
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