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Mirrors > Home > MPE Home > Th. List > cbval2 | Structured version Visualization version GIF version |
Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker cbval2v 2359 if possible. (Contributed by NM, 22-Dec-2003.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 11-Sep-2023.) (New usage is discouraged.) |
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 2338 | . 2 ⊢ Ⅎ𝑧∀𝑦𝜑 |
3 | cbval2.3 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
4 | 3 | nfal 2338 | . 2 ⊢ Ⅎ𝑥∀𝑤𝜓 |
5 | nfv 1911 | . . 3 ⊢ Ⅎ𝑦 𝑥 = 𝑧 | |
6 | nfv 1911 | . . 3 ⊢ Ⅎ𝑤 𝑥 = 𝑧 | |
7 | cbval2.2 | . . . 4 ⊢ Ⅎ𝑤𝜑 | |
8 | 7 | a1i 11 | . . 3 ⊢ (𝑥 = 𝑧 → Ⅎ𝑤𝜑) |
9 | cbval2.4 | . . . 4 ⊢ Ⅎ𝑦𝜓 | |
10 | 9 | a1i 11 | . . 3 ⊢ (𝑥 = 𝑧 → Ⅎ𝑦𝜓) |
11 | cbval2.5 | . . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) | |
12 | 11 | ex 415 | . . 3 ⊢ (𝑥 = 𝑧 → (𝑦 = 𝑤 → (𝜑 ↔ 𝜓))) |
13 | 5, 6, 8, 10, 12 | cbv2 2419 | . 2 ⊢ (𝑥 = 𝑧 → (∀𝑦𝜑 ↔ ∀𝑤𝜓)) |
14 | 2, 4, 13 | cbval 2412 | 1 ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1531 Ⅎwnf 1780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-10 2141 ax-11 2157 ax-12 2173 ax-13 2386 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1777 df-nf 1781 |
This theorem is referenced by: cbvex2 2430 |
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