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| Mirrors > Home > MPE Home > Th. List > cbvexd | Structured version Visualization version GIF version | ||
| Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2456. Usage of this theorem is discouraged because it depends on ax-13 2377. Use the weaker cbvexdw 2341 if possible. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| cbvald.1 | ⊢ Ⅎ𝑦𝜑 |
| cbvald.2 | ⊢ (𝜑 → Ⅎ𝑦𝜓) |
| cbvald.3 | ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) |
| Ref | Expression |
|---|---|
| cbvexd | ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cbvald.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
| 2 | cbvald.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑦𝜓) | |
| 3 | 2 | nfnd 1858 | . . . 4 ⊢ (𝜑 → Ⅎ𝑦 ¬ 𝜓) |
| 4 | cbvald.3 | . . . . 5 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) | |
| 5 | notbi 319 | . . . . 5 ⊢ ((𝜓 ↔ 𝜒) ↔ (¬ 𝜓 ↔ ¬ 𝜒)) | |
| 6 | 4, 5 | imbitrdi 251 | . . . 4 ⊢ (𝜑 → (𝑥 = 𝑦 → (¬ 𝜓 ↔ ¬ 𝜒))) |
| 7 | 1, 3, 6 | cbvald 2412 | . . 3 ⊢ (𝜑 → (∀𝑥 ¬ 𝜓 ↔ ∀𝑦 ¬ 𝜒)) |
| 8 | alnex 1781 | . . 3 ⊢ (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓) | |
| 9 | alnex 1781 | . . 3 ⊢ (∀𝑦 ¬ 𝜒 ↔ ¬ ∃𝑦𝜒) | |
| 10 | 7, 8, 9 | 3bitr3g 313 | . 2 ⊢ (𝜑 → (¬ ∃𝑥𝜓 ↔ ¬ ∃𝑦𝜒)) |
| 11 | 10 | con4bid 317 | 1 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∀wal 1538 ∃wex 1779 Ⅎwnf 1783 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-11 2157 ax-12 2177 ax-13 2377 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 df-nf 1784 |
| This theorem is referenced by: cbvexdva 2415 dfid3 5581 axrepndlem2 10633 axunnd 10636 axpowndlem2 10638 axpownd 10641 axregndlem2 10643 axinfndlem1 10645 axacndlem4 10650 wl-mo2df 37571 wl-eudf 37573 |
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