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| Mirrors > Home > MPE Home > Th. List > cbveu | 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 2377. Use the weaker cbveuw 2606, cbveuvw 2605 when possible. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| cbveu.1 | ⊢ Ⅎ𝑦𝜑 | 
| cbveu.2 | ⊢ Ⅎ𝑥𝜓 | 
| cbveu.3 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | 
| Ref | Expression | 
|---|---|
| cbveu | ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | cbveu.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
| 2 | 1 | sb8eu 2600 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑) | 
| 3 | cbveu.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
| 4 | cbveu.3 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
| 5 | 3, 4 | sbie 2507 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) | 
| 6 | 5 | eubii 2585 | . 2 ⊢ (∃!𝑦[𝑦 / 𝑥]𝜑 ↔ ∃!𝑦𝜓) | 
| 7 | 2, 6 | bitri 275 | 1 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 Ⅎwnf 1783 [wsb 2064 ∃!weu 2568 | 
| 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-10 2141 ax-11 2157 ax-12 2177 ax-13 2377 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 | 
| This theorem is referenced by: cbvreu 3428 cbvreucsf 3943 | 
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