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Mirrors > Home > ILE Home > Th. List > cbvexd | GIF version |
Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 1992. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof rewritten by Jim Kingdon, 10-Jun-2018.) |
Ref | Expression |
---|---|
cbvexd.1 | ⊢ Ⅎ𝑦𝜑 |
cbvexd.2 | ⊢ (𝜑 → Ⅎ𝑦𝜓) |
cbvexd.3 | ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) |
Ref | Expression |
---|---|
cbvexd | ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvexd.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | nfri 1499 | . 2 ⊢ (𝜑 → ∀𝑦𝜑) |
3 | cbvexd.2 | . . 3 ⊢ (𝜑 → Ⅎ𝑦𝜓) | |
4 | 3 | nfrd 1500 | . 2 ⊢ (𝜑 → (𝜓 → ∀𝑦𝜓)) |
5 | cbvexd.3 | . 2 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) | |
6 | 2, 4, 5 | cbvexdh 1898 | 1 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 Ⅎwnf 1436 ∃wex 1468 |
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-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 |
This theorem depends on definitions: df-bi 116 df-nf 1437 |
This theorem is referenced by: cbvexdva 1901 vtoclgft 2736 bdsepnft 13085 strcollnft 13182 |
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