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Mirrors > Home > ILE Home > Th. List > cbvex2v | GIF version |
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-Jul-1995.) |
Ref | Expression |
---|---|
cbval2v.1 | ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvex2v | ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1515 | . 2 ⊢ Ⅎ𝑧𝜑 | |
2 | nfv 1515 | . 2 ⊢ Ⅎ𝑤𝜑 | |
3 | nfv 1515 | . 2 ⊢ Ⅎ𝑥𝜓 | |
4 | nfv 1515 | . 2 ⊢ Ⅎ𝑦𝜓 | |
5 | cbval2v.1 | . 2 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) | |
6 | 1, 2, 3, 4, 5 | cbvex2 1909 | 1 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∃wex 1479 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 |
This theorem depends on definitions: df-bi 116 df-nf 1448 |
This theorem is referenced by: cbvex4v 1917 th3qlem1 6595 |
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