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Mirrors > Home > ILE Home > Th. List > cbv2h | GIF version |
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
cbv2h.1 | ⊢ (𝜑 → (𝜓 → ∀𝑦𝜓)) |
cbv2h.2 | ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) |
cbv2h.3 | ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) |
Ref | Expression |
---|---|
cbv2h | ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbv2h.1 | . . 3 ⊢ (𝜑 → (𝜓 → ∀𝑦𝜓)) | |
2 | cbv2h.2 | . . 3 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) | |
3 | cbv2h.3 | . . . 4 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) | |
4 | bi1 111 | . . . 4 ⊢ ((𝜓 ↔ 𝜒) → (𝜓 → 𝜒)) | |
5 | 3, 4 | syl6 29 | . . 3 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒))) |
6 | 1, 2, 5 | cbv1h 1633 | . 2 ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒)) |
7 | equcomi 1592 | . . . . 5 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦) | |
8 | bi2 121 | . . . . 5 ⊢ ((𝜓 ↔ 𝜒) → (𝜒 → 𝜓)) | |
9 | 7, 3, 8 | syl56 30 | . . . 4 ⊢ (𝜑 → (𝑦 = 𝑥 → (𝜒 → 𝜓))) |
10 | 2, 1, 9 | cbv1h 1633 | . . 3 ⊢ (∀𝑦∀𝑥𝜑 → (∀𝑦𝜒 → ∀𝑥𝜓)) |
11 | 10 | a7s 1343 | . 2 ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑦𝜒 → ∀𝑥𝜓)) |
12 | 6, 11 | impbid 120 | 1 ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∀wal 1241 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 |
This theorem depends on definitions: df-bi 110 df-nf 1350 |
This theorem is referenced by: cbv2 1635 |
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