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Mirrors > Home > ILE Home > Th. List > cbval2 | GIF version |
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 22-Dec-2003.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 22-Apr-2018.) |
Ref | Expression |
---|---|
cbval2.1 | ⊢ Ⅎ𝑧𝜑 |
cbval2.2 | ⊢ Ⅎ𝑤𝜑 |
cbval2.3 | ⊢ Ⅎ𝑥𝜓 |
cbval2.4 | ⊢ Ⅎ𝑦𝜓 |
cbval2.5 | ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbval2 | ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbval2.1 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
2 | 1 | nfal 1556 | . 2 ⊢ Ⅎ𝑧∀𝑦𝜑 |
3 | cbval2.3 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
4 | 3 | nfal 1556 | . 2 ⊢ Ⅎ𝑥∀𝑤𝜓 |
5 | nfv 1508 | . . . . . 6 ⊢ Ⅎ𝑤 𝑥 = 𝑧 | |
6 | cbval2.2 | . . . . . 6 ⊢ Ⅎ𝑤𝜑 | |
7 | 5, 6 | nfim 1552 | . . . . 5 ⊢ Ⅎ𝑤(𝑥 = 𝑧 → 𝜑) |
8 | nfv 1508 | . . . . . 6 ⊢ Ⅎ𝑦 𝑥 = 𝑧 | |
9 | cbval2.4 | . . . . . 6 ⊢ Ⅎ𝑦𝜓 | |
10 | 8, 9 | nfim 1552 | . . . . 5 ⊢ Ⅎ𝑦(𝑥 = 𝑧 → 𝜓) |
11 | cbval2.5 | . . . . . . 7 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) | |
12 | 11 | expcom 115 | . . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑥 = 𝑧 → (𝜑 ↔ 𝜓))) |
13 | 12 | pm5.74d 181 | . . . . 5 ⊢ (𝑦 = 𝑤 → ((𝑥 = 𝑧 → 𝜑) ↔ (𝑥 = 𝑧 → 𝜓))) |
14 | 7, 10, 13 | cbval 1734 | . . . 4 ⊢ (∀𝑦(𝑥 = 𝑧 → 𝜑) ↔ ∀𝑤(𝑥 = 𝑧 → 𝜓)) |
15 | 19.21v 1853 | . . . 4 ⊢ (∀𝑦(𝑥 = 𝑧 → 𝜑) ↔ (𝑥 = 𝑧 → ∀𝑦𝜑)) | |
16 | 19.21v 1853 | . . . 4 ⊢ (∀𝑤(𝑥 = 𝑧 → 𝜓) ↔ (𝑥 = 𝑧 → ∀𝑤𝜓)) | |
17 | 14, 15, 16 | 3bitr3i 209 | . . 3 ⊢ ((𝑥 = 𝑧 → ∀𝑦𝜑) ↔ (𝑥 = 𝑧 → ∀𝑤𝜓)) |
18 | 17 | pm5.74ri 180 | . 2 ⊢ (𝑥 = 𝑧 → (∀𝑦𝜑 ↔ ∀𝑤𝜓)) |
19 | 2, 4, 18 | cbval 1734 | 1 ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∀wal 1333 Ⅎwnf 1440 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 |
This theorem depends on definitions: df-bi 116 df-nf 1441 |
This theorem is referenced by: cbval2v 1903 |
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