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Mirrors > Home > MPE Home > Th. List > 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 3896 | . . 3 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌(𝐵 ∘ 𝐶) = ⦋𝐴 / 𝑥⦌(𝐵 ∘ 𝐶)) | |
2 | csbeq1 3896 | . . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵) | |
3 | csbeq1 3896 | . . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐶) | |
4 | 2, 3 | coeq12d 5864 | . . 3 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌𝐵 ∘ ⦋𝑦 / 𝑥⦌𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∘ ⦋𝐴 / 𝑥⦌𝐶)) |
5 | 1, 4 | eqeq12d 2747 | . 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 3477 | . . 3 ⊢ 𝑦 ∈ V | |
7 | nfcsb1v 3918 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | |
8 | nfcsb1v 3918 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶 | |
9 | 7, 8 | nfco 5865 | . . 3 ⊢ Ⅎ𝑥(⦋𝑦 / 𝑥⦌𝐵 ∘ ⦋𝑦 / 𝑥⦌𝐶) |
10 | csbeq1a 3907 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
11 | csbeq1a 3907 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶) | |
12 | 10, 11 | coeq12d 5864 | . . 3 ⊢ (𝑥 = 𝑦 → (𝐵 ∘ 𝐶) = (⦋𝑦 / 𝑥⦌𝐵 ∘ ⦋𝑦 / 𝑥⦌𝐶)) |
13 | 6, 9, 12 | csbief 3928 | . 2 ⊢ ⦋𝑦 / 𝑥⦌(𝐵 ∘ 𝐶) = (⦋𝑦 / 𝑥⦌𝐵 ∘ ⦋𝑦 / 𝑥⦌𝐶) |
14 | 5, 13 | vtoclg 3542 | 1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵 ∘ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∘ ⦋𝐴 / 𝑥⦌𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 ⦋csb 3893 ∘ ccom 5680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-co 5685 |
This theorem is referenced by: csbwrecsg 8312 brtrclfv2 42793 |
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