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Mirrors > Home > ILE Home > Th. List > cbveu | 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 2039 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑) |
3 | cbveu.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
4 | cbveu.3 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
5 | 3, 4 | sbie 1791 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
6 | 5 | eubii 2035 | . 2 ⊢ (∃!𝑦[𝑦 / 𝑥]𝜑 ↔ ∃!𝑦𝜓) |
7 | 2, 6 | bitri 184 | 1 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 Ⅎwnf 1460 [wsb 1762 ∃!weu 2026 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 |
This theorem is referenced by: cbvmo 2066 cbvreu 2701 cbvreucsf 3121 tz6.12f 5544 f1ompt 5667 climeu 11299 |
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