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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 2891 with a disjoint variable condition, which does not require ax-13 2390. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Gino Giotto, 10-Jan-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 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | sbco2v 2352 | . . . 4 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑) |
3 | cbvabw.2 | . . . . . 6 ⊢ Ⅎ𝑥𝜓 | |
4 | cbvabw.3 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
5 | 3, 4 | sbiev 2330 | . . . . 5 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
6 | 5 | sbbii 2081 | . . . 4 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓) |
7 | 2, 6 | bitr3i 279 | . . 3 ⊢ ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓) |
8 | df-clab 2800 | . . 3 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ [𝑧 / 𝑥]𝜑) | |
9 | df-clab 2800 | . . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ [𝑧 / 𝑦]𝜓) | |
10 | 7, 8, 9 | 3bitr4i 305 | . 2 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ 𝑧 ∈ {𝑦 ∣ 𝜓}) |
11 | 10 | eqriv 2818 | 1 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 Ⅎwnf 1784 [wsb 2069 ∈ wcel 2114 {cab 2799 |
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 1911 ax-6 1970 ax-7 2015 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 |
This theorem is referenced by: cbvrabw 3489 cbvsbcw 3804 cbvrabcsfw 3924 rabsnifsb 4658 dfdmf 5765 dfrnf 5820 funfv2f 6752 abrexex2g 7665 bnj873 32196 ptrest 34906 poimirlem26 34933 poimirlem27 34934 |
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