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Mirrors > Home > MPE Home > Th. List > cbvcsbv | Structured version Visualization version GIF version |
Description: Change the bound variable of a proper substitution into a class using implicit substitution. (Contributed by NM, 30-Sep-2008.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Ref | Expression |
---|---|
cbvcsbv.1 | ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
cbvcsbv | ⊢ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑦⦌𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvcsbv.1 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
2 | 1 | eleq2d 2825 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶)) |
3 | 2 | cbvsbcvw 3826 | . . 3 ⊢ ([𝐴 / 𝑥]𝑧 ∈ 𝐵 ↔ [𝐴 / 𝑦]𝑧 ∈ 𝐶) |
4 | 3 | abbii 2807 | . 2 ⊢ {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐵} = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐶} |
5 | df-csb 3909 | . 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐵} | |
6 | df-csb 3909 | . 2 ⊢ ⦋𝐴 / 𝑦⦌𝐶 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐶} | |
7 | 4, 5, 6 | 3eqtr4i 2773 | 1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑦⦌𝐶 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 {cab 2712 [wsbc 3791 ⦋csb 3908 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-sbc 3792 df-csb 3909 |
This theorem is referenced by: cbvsumv 15729 cbvprodv 15947 pmatcollpw3lem 22805 precsexlemcbv 28245 cbvprodvw2 36230 poimirlem27 37634 cdleme40v 40452 |
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