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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. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) |
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 2503 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑) |
3 | cbveu.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
4 | cbveu.3 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
5 | 3, 4 | sbie 2408 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
6 | 5 | eubii 2492 | . 2 ⊢ (∃!𝑦[𝑦 / 𝑥]𝜑 ↔ ∃!𝑦𝜓) |
7 | 2, 6 | bitri 264 | 1 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 Ⅎwnf 1708 [wsb 1880 ∃!weu 2470 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 |
This theorem is referenced by: cbvmo 2506 cbvreu 3169 cbvreucsf 3567 tz6.12f 6212 f1ompt 6382 climeu 14286 initoeu2 16666 |
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