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Mirrors > Home > MPE Home > Th. List > cbvsbcvw | Structured version Visualization version GIF version |
Description: Change the bound variable of a class substitution using implicit substitution. Version of cbvsbcv 3814 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by NM, 30-Sep-2008.) Avoid ax-13 2370. (Revised by Gino Giotto, 10-Jan-2024.) |
Ref | Expression |
---|---|
cbvsbcvw.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvsbcvw | ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvsbcvw.1 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
2 | 1 | cbvabv 2804 | . . 3 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} |
3 | 2 | eleq2i 2824 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝐴 ∈ {𝑦 ∣ 𝜓}) |
4 | df-sbc 3778 | . 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) | |
5 | df-sbc 3778 | . 2 ⊢ ([𝐴 / 𝑦]𝜓 ↔ 𝐴 ∈ {𝑦 ∣ 𝜓}) | |
6 | 3, 4, 5 | 3bitr4i 303 | 1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2105 {cab 2708 [wsbc 3777 |
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-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-sbc 3778 |
This theorem is referenced by: frpoins3xpg 8129 frpoins3xp3g 8130 fpwwe2cbv 10628 reuf1odnf 46114 |
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