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Mirrors > Home > MPE Home > Th. List > cbvsbcw | Structured version Visualization version GIF version |
Description: Change bound variables in a wff substitution. Version of cbvsbc 3763 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by Jeff Hankins, 19-Sep-2009.) Avoid ax-13 2370. (Revised by Gino Giotto, 10-Jan-2024.) |
Ref | Expression |
---|---|
cbvsbcw.1 | ⊢ Ⅎ𝑦𝜑 |
cbvsbcw.2 | ⊢ Ⅎ𝑥𝜓 |
cbvsbcw.3 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvsbcw | ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvsbcw.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
2 | cbvsbcw.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
3 | cbvsbcw.3 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | cbvabw 2810 | . . 3 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} |
5 | 4 | eleq2i 2828 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝐴 ∈ {𝑦 ∣ 𝜓}) |
6 | df-sbc 3728 | . 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) | |
7 | df-sbc 3728 | . 2 ⊢ ([𝐴 / 𝑦]𝜓 ↔ 𝐴 ∈ {𝑦 ∣ 𝜓}) | |
8 | 5, 6, 7 | 3bitr4i 302 | 1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 Ⅎwnf 1784 ∈ wcel 2105 {cab 2713 [wsbc 3727 |
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-11 2153 ax-12 2170 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1781 df-nf 1785 df-sb 2067 df-clab 2714 df-cleq 2728 df-clel 2814 df-sbc 3728 |
This theorem is referenced by: cbvcsbw 3853 |
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