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Mirrors > Home > ILE Home > Th. List > cbvmo | GIF version |
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 9-Mar-1995.) (Revised by Andrew Salmon, 8-Jun-2011.) |
Ref | Expression |
---|---|
cbvmo.1 | ⊢ Ⅎ𝑦𝜑 |
cbvmo.2 | ⊢ Ⅎ𝑥𝜓 |
cbvmo.3 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvmo | ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvmo.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
2 | cbvmo.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
3 | cbvmo.3 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | cbvex 1729 | . . 3 ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓) |
5 | 1, 2, 3 | cbveu 2023 | . . 3 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓) |
6 | 4, 5 | imbi12i 238 | . 2 ⊢ ((∃𝑥𝜑 → ∃!𝑥𝜑) ↔ (∃𝑦𝜓 → ∃!𝑦𝜓)) |
7 | df-mo 2003 | . 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) | |
8 | df-mo 2003 | . 2 ⊢ (∃*𝑦𝜓 ↔ (∃𝑦𝜓 → ∃!𝑦𝜓)) | |
9 | 6, 7, 8 | 3bitr4i 211 | 1 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 Ⅎwnf 1436 ∃wex 1468 ∃!weu 1999 ∃*wmo 2000 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 |
This theorem is referenced by: dffun6f 5136 |
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