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Mirrors > Home > MPE Home > Th. List > cbval2v | Structured version Visualization version GIF version |
Description: Rule used to change bound variables, using implicit substitution. Version of cbval2 2411 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by NM, 22-Dec-2003.) (Revised by BJ, 16-Jun-2019.) (Proof shortened by Gino Giotto, 10-Jan-2024.) |
Ref | Expression |
---|---|
cbval2v.1 | ⊢ Ⅎ𝑧𝜑 |
cbval2v.2 | ⊢ Ⅎ𝑤𝜑 |
cbval2v.3 | ⊢ Ⅎ𝑥𝜓 |
cbval2v.4 | ⊢ Ⅎ𝑦𝜓 |
cbval2v.5 | ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbval2v | ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbval2v.1 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
2 | 1 | nfal 2317 | . 2 ⊢ Ⅎ𝑧∀𝑦𝜑 |
3 | cbval2v.3 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
4 | 3 | nfal 2317 | . 2 ⊢ Ⅎ𝑥∀𝑤𝜓 |
5 | nfv 1917 | . . 3 ⊢ Ⅎ𝑦 𝑥 = 𝑧 | |
6 | nfv 1917 | . . 3 ⊢ Ⅎ𝑤 𝑥 = 𝑧 | |
7 | cbval2v.2 | . . . 4 ⊢ Ⅎ𝑤𝜑 | |
8 | 7 | a1i 11 | . . 3 ⊢ (𝑥 = 𝑧 → Ⅎ𝑤𝜑) |
9 | cbval2v.4 | . . . 4 ⊢ Ⅎ𝑦𝜓 | |
10 | 9 | a1i 11 | . . 3 ⊢ (𝑥 = 𝑧 → Ⅎ𝑦𝜓) |
11 | cbval2v.5 | . . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) | |
12 | 11 | ex 413 | . . 3 ⊢ (𝑥 = 𝑧 → (𝑦 = 𝑤 → (𝜑 ↔ 𝜓))) |
13 | 5, 6, 8, 10, 12 | cbv2w 2334 | . 2 ⊢ (𝑥 = 𝑧 → (∀𝑦𝜑 ↔ ∀𝑤𝜓)) |
14 | 2, 4, 13 | cbvalv1 2338 | 1 ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∀wal 1537 Ⅎwnf 1786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-10 2137 ax-11 2154 ax-12 2171 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-ex 1783 df-nf 1787 |
This theorem is referenced by: cbvex2v 2342 bj-cbval2vv 34983 eqrelf 36395 |
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