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| Mirrors > Home > MPE Home > Th. List > cbvab | Structured version Visualization version GIF version | ||
| Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2376. Usage of the weaker cbvabw 2812 and cbvabv 2811 are preferred. (Contributed by Andrew Salmon, 11-Jul-2011.) (Proof shortened by Wolf Lammen, 16-Nov-2019.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| cbvab.1 | ⊢ Ⅎ𝑦𝜑 | 
| cbvab.2 | ⊢ Ⅎ𝑥𝜓 | 
| cbvab.3 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | 
| Ref | Expression | 
|---|---|
| cbvab | ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | cbvab.1 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
| 2 | 1 | sbco2 2515 | . . . 4 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑) | 
| 3 | cbvab.2 | . . . . . 6 ⊢ Ⅎ𝑥𝜓 | |
| 4 | cbvab.3 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
| 5 | 3, 4 | sbie 2506 | . . . . 5 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) | 
| 6 | 5 | sbbii 2075 | . . . 4 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓) | 
| 7 | 2, 6 | bitr3i 277 | . . 3 ⊢ ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓) | 
| 8 | df-clab 2714 | . . 3 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ [𝑧 / 𝑥]𝜑) | |
| 9 | df-clab 2714 | . . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ [𝑧 / 𝑦]𝜓) | |
| 10 | 7, 8, 9 | 3bitr4i 303 | . 2 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ 𝑧 ∈ {𝑦 ∣ 𝜓}) | 
| 11 | 10 | eqriv 2733 | 1 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1539 Ⅎwnf 1782 [wsb 2063 ∈ wcel 2107 {cab 2713 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-13 2376 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 | 
| This theorem is referenced by: cbvrab 3478 cdeqab1 3777 cbvsbc 3822 cbvrabcsf 3943 | 
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