Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > cbvex2 | 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 2373. Use the weaker cbvex2v 2345 if possible. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 16-Jun-2019.) (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 |
---|---|
cbvex2 | ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbval2.1 | . . . . 5 ⊢ Ⅎ𝑧𝜑 | |
2 | 1 | nfn 1863 | . . . 4 ⊢ Ⅎ𝑧 ¬ 𝜑 |
3 | cbval2.2 | . . . . 5 ⊢ Ⅎ𝑤𝜑 | |
4 | 3 | nfn 1863 | . . . 4 ⊢ Ⅎ𝑤 ¬ 𝜑 |
5 | cbval2.3 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
6 | 5 | nfn 1863 | . . . 4 ⊢ Ⅎ𝑥 ¬ 𝜓 |
7 | cbval2.4 | . . . . 5 ⊢ Ⅎ𝑦𝜓 | |
8 | 7 | nfn 1863 | . . . 4 ⊢ Ⅎ𝑦 ¬ 𝜓 |
9 | cbval2.5 | . . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) | |
10 | 9 | notbid 317 | . . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (¬ 𝜑 ↔ ¬ 𝜓)) |
11 | 2, 4, 6, 8, 10 | cbval2 2412 | . . 3 ⊢ (∀𝑥∀𝑦 ¬ 𝜑 ↔ ∀𝑧∀𝑤 ¬ 𝜓) |
12 | 2nexaln 1835 | . . 3 ⊢ (¬ ∃𝑥∃𝑦𝜑 ↔ ∀𝑥∀𝑦 ¬ 𝜑) | |
13 | 2nexaln 1835 | . . 3 ⊢ (¬ ∃𝑧∃𝑤𝜓 ↔ ∀𝑧∀𝑤 ¬ 𝜓) | |
14 | 11, 12, 13 | 3bitr4i 302 | . 2 ⊢ (¬ ∃𝑥∃𝑦𝜑 ↔ ¬ ∃𝑧∃𝑤𝜓) |
15 | 14 | con4bii 320 | 1 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1539 ∃wex 1785 Ⅎwnf 1789 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-10 2140 ax-11 2157 ax-12 2174 ax-13 2373 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-ex 1786 df-nf 1790 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |