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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 2430 with a disjoint variable condition, which does not require ax-13 2386. (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 1853 | . . . 4 ⊢ Ⅎ𝑧 ¬ 𝜑 |
3 | cbval2v.2 | . . . . 5 ⊢ Ⅎ𝑤𝜑 | |
4 | 3 | nfn 1853 | . . . 4 ⊢ Ⅎ𝑤 ¬ 𝜑 |
5 | cbval2v.3 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
6 | 5 | nfn 1853 | . . . 4 ⊢ Ⅎ𝑥 ¬ 𝜓 |
7 | cbval2v.4 | . . . . 5 ⊢ Ⅎ𝑦𝜓 | |
8 | 7 | nfn 1853 | . . . 4 ⊢ Ⅎ𝑦 ¬ 𝜓 |
9 | cbval2v.5 | . . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) | |
10 | 9 | notbid 320 | . . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (¬ 𝜑 ↔ ¬ 𝜓)) |
11 | 2, 4, 6, 8, 10 | cbval2v 2359 | . . 3 ⊢ (∀𝑥∀𝑦 ¬ 𝜑 ↔ ∀𝑧∀𝑤 ¬ 𝜓) |
12 | 11 | notbii 322 | . 2 ⊢ (¬ ∀𝑥∀𝑦 ¬ 𝜑 ↔ ¬ ∀𝑧∀𝑤 ¬ 𝜓) |
13 | 2exnaln 1825 | . 2 ⊢ (∃𝑥∃𝑦𝜑 ↔ ¬ ∀𝑥∀𝑦 ¬ 𝜑) | |
14 | 2exnaln 1825 | . 2 ⊢ (∃𝑧∃𝑤𝜓 ↔ ¬ ∀𝑧∀𝑤 ¬ 𝜓) | |
15 | 12, 13, 14 | 3bitr4i 305 | 1 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1531 ∃wex 1776 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 |
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: cbvopab 5136 cbvoprab12 7242 bj-cbvex2vv 34124 or2expropbilem2 43267 ichnreuop 43633 |
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