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Mirrors > Home > MPE Home > Th. List > cbvabw | Structured version Visualization version GIF version |
Description: Rule used to change bound variables, using implicit substitution. Version of cbvab 2812 with a disjoint variable condition, which does not require ax-10 2139, ax-13 2375. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by GG, 23-May-2024.) |
Ref | Expression |
---|---|
cbvabw.1 | ⊢ Ⅎ𝑦𝜑 |
cbvabw.2 | ⊢ Ⅎ𝑥𝜓 |
cbvabw.3 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvabw | ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvabw.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
2 | cbvabw.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
3 | cbvabw.3 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | cbvsbvf 2364 | . . 3 ⊢ ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓) |
5 | df-clab 2713 | . . 3 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ [𝑧 / 𝑥]𝜑) | |
6 | df-clab 2713 | . . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ [𝑧 / 𝑦]𝜓) | |
7 | 4, 5, 6 | 3bitr4i 303 | . 2 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ 𝑧 ∈ {𝑦 ∣ 𝜓}) |
8 | 7 | eqriv 2732 | 1 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 Ⅎwnf 1780 [wsb 2062 ∈ wcel 2106 {cab 2712 |
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-9 2116 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 |
This theorem is referenced by: cbvrabw 3471 cbvrabwOLD 3472 cbvsbcw 3825 cbvrabcsfw 3952 rabsnifsb 4727 dfdmf 5910 dfrnf 5964 funfv2f 6998 abrexex2g 7988 bnj873 34917 fineqvrep 35088 ptrest 37606 poimirlem26 37633 poimirlem27 37634 |
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