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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. (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 2153 | . 2 ⊢ Ⅎ𝑧∀𝑦𝜑 |
3 | cbval2.3 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
4 | 3 | nfal 2153 | . 2 ⊢ Ⅎ𝑥∀𝑤𝜓 |
5 | nfv 1843 | . . . . . 6 ⊢ Ⅎ𝑤 𝑥 = 𝑧 | |
6 | cbval2.2 | . . . . . 6 ⊢ Ⅎ𝑤𝜑 | |
7 | 5, 6 | nfim 1825 | . . . . 5 ⊢ Ⅎ𝑤(𝑥 = 𝑧 → 𝜑) |
8 | nfv 1843 | . . . . . 6 ⊢ Ⅎ𝑦 𝑥 = 𝑧 | |
9 | cbval2.4 | . . . . . 6 ⊢ Ⅎ𝑦𝜓 | |
10 | 8, 9 | nfim 1825 | . . . . 5 ⊢ Ⅎ𝑦(𝑥 = 𝑧 → 𝜓) |
11 | cbval2.5 | . . . . . . 7 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) | |
12 | 11 | expcom 451 | . . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑥 = 𝑧 → (𝜑 ↔ 𝜓))) |
13 | 12 | pm5.74d 262 | . . . . 5 ⊢ (𝑦 = 𝑤 → ((𝑥 = 𝑧 → 𝜑) ↔ (𝑥 = 𝑧 → 𝜓))) |
14 | 7, 10, 13 | cbval 2271 | . . . 4 ⊢ (∀𝑦(𝑥 = 𝑧 → 𝜑) ↔ ∀𝑤(𝑥 = 𝑧 → 𝜓)) |
15 | 19.21v 1868 | . . . 4 ⊢ (∀𝑦(𝑥 = 𝑧 → 𝜑) ↔ (𝑥 = 𝑧 → ∀𝑦𝜑)) | |
16 | 19.21v 1868 | . . . 4 ⊢ (∀𝑤(𝑥 = 𝑧 → 𝜓) ↔ (𝑥 = 𝑧 → ∀𝑤𝜓)) | |
17 | 14, 15, 16 | 3bitr3i 290 | . . 3 ⊢ ((𝑥 = 𝑧 → ∀𝑦𝜑) ↔ (𝑥 = 𝑧 → ∀𝑤𝜓)) |
18 | 17 | pm5.74ri 261 | . 2 ⊢ (𝑥 = 𝑧 → (∀𝑦𝜑 ↔ ∀𝑤𝜓)) |
19 | 2, 4, 18 | cbval 2271 | 1 ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∀wal 1481 Ⅎwnf 1708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 df-nf 1710 |
This theorem is referenced by: cbvex2 2280 cbval2vOLD 2286 eqrelf 34020 |
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