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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 3502 | . . 3 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌(𝐵 ∘ 𝐶) = ⦋𝐴 / 𝑥⦌(𝐵 ∘ 𝐶)) | |
2 | csbeq1 3502 | . . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵) | |
3 | csbeq1 3502 | . . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐶) | |
4 | 2, 3 | coeq12d 5208 | . . 3 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌𝐵 ∘ ⦋𝑦 / 𝑥⦌𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∘ ⦋𝐴 / 𝑥⦌𝐶)) |
5 | 1, 4 | eqeq12d 2625 | . 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 3176 | . . 3 ⊢ 𝑦 ∈ V | |
7 | nfcsb1v 3515 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | |
8 | nfcsb1v 3515 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶 | |
9 | 7, 8 | nfco 5209 | . . 3 ⊢ Ⅎ𝑥(⦋𝑦 / 𝑥⦌𝐵 ∘ ⦋𝑦 / 𝑥⦌𝐶) |
10 | csbeq1a 3508 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
11 | csbeq1a 3508 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶) | |
12 | 10, 11 | coeq12d 5208 | . . 3 ⊢ (𝑥 = 𝑦 → (𝐵 ∘ 𝐶) = (⦋𝑦 / 𝑥⦌𝐵 ∘ ⦋𝑦 / 𝑥⦌𝐶)) |
13 | 6, 9, 12 | csbief 3524 | . 2 ⊢ ⦋𝑦 / 𝑥⦌(𝐵 ∘ 𝐶) = (⦋𝑦 / 𝑥⦌𝐵 ∘ ⦋𝑦 / 𝑥⦌𝐶) |
14 | 5, 13 | vtoclg 3239 | 1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵 ∘ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∘ ⦋𝐴 / 𝑥⦌𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ⦋csb 3499 ∘ ccom 5042 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-co 5047 |
This theorem is referenced by: brtrclfv2 37038 |
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