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Mirrors > Home > MPE Home > Th. List > cbvex2v | Structured version Visualization version GIF version |
Description: Rule used to change bound variables, using implicit substitution. Version of cbvex2 2423 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by NM, 14-Sep-2003.) (Revised by BJ, 16-Jun-2019.) |
Ref | Expression |
---|---|
cbval2v.1 | ⊢ Ⅎ𝑧𝜑 |
cbval2v.2 | ⊢ Ⅎ𝑤𝜑 |
cbval2v.3 | ⊢ Ⅎ𝑥𝜓 |
cbval2v.4 | ⊢ Ⅎ𝑦𝜓 |
cbval2v.5 | ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvex2v | ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbval2v.1 | . . . . 5 ⊢ Ⅎ𝑧𝜑 | |
2 | 1 | nfn 1858 | . . . 4 ⊢ Ⅎ𝑧 ¬ 𝜑 |
3 | cbval2v.2 | . . . . 5 ⊢ Ⅎ𝑤𝜑 | |
4 | 3 | nfn 1858 | . . . 4 ⊢ Ⅎ𝑤 ¬ 𝜑 |
5 | cbval2v.3 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
6 | 5 | nfn 1858 | . . . 4 ⊢ Ⅎ𝑥 ¬ 𝜓 |
7 | cbval2v.4 | . . . . 5 ⊢ Ⅎ𝑦𝜓 | |
8 | 7 | nfn 1858 | . . . 4 ⊢ Ⅎ𝑦 ¬ 𝜓 |
9 | cbval2v.5 | . . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) | |
10 | 9 | notbid 321 | . . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (¬ 𝜑 ↔ ¬ 𝜓)) |
11 | 2, 4, 6, 8, 10 | cbval2v 2352 | . . 3 ⊢ (∀𝑥∀𝑦 ¬ 𝜑 ↔ ∀𝑧∀𝑤 ¬ 𝜓) |
12 | 11 | notbii 323 | . 2 ⊢ (¬ ∀𝑥∀𝑦 ¬ 𝜑 ↔ ¬ ∀𝑧∀𝑤 ¬ 𝜓) |
13 | 2exnaln 1830 | . 2 ⊢ (∃𝑥∃𝑦𝜑 ↔ ¬ ∀𝑥∀𝑦 ¬ 𝜑) | |
14 | 2exnaln 1830 | . 2 ⊢ (∃𝑧∃𝑤𝜓 ↔ ¬ ∀𝑧∀𝑤 ¬ 𝜓) | |
15 | 12, 13, 14 | 3bitr4i 306 | 1 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1536 ∃wex 1781 Ⅎwnf 1785 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2142 ax-11 2158 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ex 1782 df-nf 1786 |
This theorem is referenced by: cbvopab 5101 cbvoprab12 7222 bj-cbvex2vv 34239 or2expropbilem2 43625 ichnreuop 43989 |
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