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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 2337. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) |
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 1785 | . . . 4 ⊢ (𝜑 → Ⅎ𝑦 ¬ 𝜓) |
4 | cbvald.3 | . . . . 5 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) | |
5 | notbi 309 | . . . . 5 ⊢ ((𝜓 ↔ 𝜒) ↔ (¬ 𝜓 ↔ ¬ 𝜒)) | |
6 | 4, 5 | syl6ib 241 | . . . 4 ⊢ (𝜑 → (𝑥 = 𝑦 → (¬ 𝜓 ↔ ¬ 𝜒))) |
7 | 1, 3, 6 | cbvald 2277 | . . 3 ⊢ (𝜑 → (∀𝑥 ¬ 𝜓 ↔ ∀𝑦 ¬ 𝜒)) |
8 | 7 | notbid 308 | . 2 ⊢ (𝜑 → (¬ ∀𝑥 ¬ 𝜓 ↔ ¬ ∀𝑦 ¬ 𝜒)) |
9 | df-ex 1705 | . 2 ⊢ (∃𝑥𝜓 ↔ ¬ ∀𝑥 ¬ 𝜓) | |
10 | df-ex 1705 | . 2 ⊢ (∃𝑦𝜒 ↔ ¬ ∀𝑦 ¬ 𝜒) | |
11 | 8, 9, 10 | 3bitr4g 303 | 1 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∀wal 1481 ∃wex 1704 Ⅎwnf 1708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-ex 1705 df-nf 1710 |
This theorem is referenced by: cbvexdvaOLD 2284 vtoclgft 3254 vtoclgftOLD 3255 dfid3 5025 axrepndlem2 9415 axunnd 9418 axpowndlem2 9420 axpownd 9423 axregndlem2 9425 axinfndlem1 9427 axacndlem4 9432 wl-mo2df 33352 wl-eudf 33354 wl-mo2t 33357 |
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