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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 2814 with a disjoint variable condition, which does not require ax-10 2141, ax-13 2377. (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 2366 | . . 3 ⊢ ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓) |
| 5 | df-clab 2715 | . . 3 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ [𝑧 / 𝑥]𝜑) | |
| 6 | df-clab 2715 | . . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ [𝑧 / 𝑦]𝜓) | |
| 7 | 4, 5, 6 | 3bitr4i 303 | . 2 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ 𝑧 ∈ {𝑦 ∣ 𝜓}) |
| 8 | 7 | eqriv 2734 | 1 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 Ⅎwnf 1783 [wsb 2064 ∈ wcel 2108 {cab 2714 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 |
| This theorem is referenced by: cbvrabw 3473 cbvrabwOLD 3474 cbvsbcw 3821 cbvrabcsfw 3940 rabsnifsb 4722 dfdmf 5907 dfrnf 5961 funfv2f 6998 abrexex2g 7989 bnj873 34938 fineqvrep 35109 ptrest 37626 poimirlem26 37653 poimirlem27 37654 |
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