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Mirrors > Home > ILE Home > Th. List > cbv1v | GIF version |
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 16-Jun-2019.) |
Ref | Expression |
---|---|
cbv1v.1 | ⊢ Ⅎ𝑥𝜑 |
cbv1v.2 | ⊢ Ⅎ𝑦𝜑 |
cbv1v.3 | ⊢ (𝜑 → Ⅎ𝑦𝜓) |
cbv1v.4 | ⊢ (𝜑 → Ⅎ𝑥𝜒) |
cbv1v.5 | ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒))) |
Ref | Expression |
---|---|
cbv1v | ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbv1v.2 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
2 | cbv1v.3 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑦𝜓) | |
3 | 1, 2 | nfim1 1558 | . . . 4 ⊢ Ⅎ𝑦(𝜑 → 𝜓) |
4 | cbv1v.1 | . . . . 5 ⊢ Ⅎ𝑥𝜑 | |
5 | cbv1v.4 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
6 | 4, 5 | nfim1 1558 | . . . 4 ⊢ Ⅎ𝑥(𝜑 → 𝜒) |
7 | cbv1v.5 | . . . . . 6 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒))) | |
8 | 7 | com12 30 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 → (𝜓 → 𝜒))) |
9 | 8 | a2d 26 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝜓) → (𝜑 → 𝜒))) |
10 | 3, 6, 9 | cbv3v 1731 | . . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → ∀𝑦(𝜑 → 𝜒)) |
11 | 4 | 19.21 1570 | . . 3 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)) |
12 | 1 | 19.21 1570 | . . 3 ⊢ (∀𝑦(𝜑 → 𝜒) ↔ (𝜑 → ∀𝑦𝜒)) |
13 | 10, 11, 12 | 3imtr3i 199 | . 2 ⊢ ((𝜑 → ∀𝑥𝜓) → (𝜑 → ∀𝑦𝜒)) |
14 | 13 | pm2.86i 98 | 1 ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1340 Ⅎwnf 1447 |
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-4 1497 ax-i9 1517 ax-ial 1521 ax-i5r 1522 |
This theorem depends on definitions: df-bi 116 df-nf 1448 |
This theorem is referenced by: cbv2w 1737 |
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