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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-sb8et | Structured version Visualization version GIF version |
Description: Substitution of variable in universal quantifier. Closed form of sb8e 2425. (Contributed by Wolf Lammen, 27-Jul-2019.) |
Ref | Expression |
---|---|
wl-sb8et | ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wl-nfnbi 33314 | . . . . 5 ⊢ (Ⅎ𝑦𝜑 ↔ Ⅎ𝑦 ¬ 𝜑) | |
2 | 1 | albii 1747 | . . . 4 ⊢ (∀𝑥Ⅎ𝑦𝜑 ↔ ∀𝑥Ⅎ𝑦 ¬ 𝜑) |
3 | wl-sb8t 33333 | . . . 4 ⊢ (∀𝑥Ⅎ𝑦 ¬ 𝜑 → (∀𝑥 ¬ 𝜑 ↔ ∀𝑦[𝑦 / 𝑥] ¬ 𝜑)) | |
4 | 2, 3 | sylbi 207 | . . 3 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥 ¬ 𝜑 ↔ ∀𝑦[𝑦 / 𝑥] ¬ 𝜑)) |
5 | alnex 1706 | . . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑) | |
6 | sbn 2391 | . . . . 5 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑) | |
7 | 6 | albii 1747 | . . . 4 ⊢ (∀𝑦[𝑦 / 𝑥] ¬ 𝜑 ↔ ∀𝑦 ¬ [𝑦 / 𝑥]𝜑) |
8 | alnex 1706 | . . . 4 ⊢ (∀𝑦 ¬ [𝑦 / 𝑥]𝜑 ↔ ¬ ∃𝑦[𝑦 / 𝑥]𝜑) | |
9 | 7, 8 | bitri 264 | . . 3 ⊢ (∀𝑦[𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ ∃𝑦[𝑦 / 𝑥]𝜑) |
10 | 4, 5, 9 | 3bitr3g 302 | . 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (¬ ∃𝑥𝜑 ↔ ¬ ∃𝑦[𝑦 / 𝑥]𝜑)) |
11 | 10 | con4bid 307 | 1 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∀wal 1481 ∃wex 1704 Ⅎwnf 1708 [wsb 1880 |
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 df-sb 1881 |
This theorem is referenced by: wl-mo3t 33358 wl-sb8mot 33360 |
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