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Mathbox for Wolf Lammen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-sb8ft | Structured version Visualization version GIF version |
Description: Substitution of variable in universal quantifier. Closed form of sb8f 2352. (Contributed by Wolf Lammen, 27-Apr-2025.) |
Ref | Expression |
---|---|
wl-sb8ft | ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbft 2266 | . . . 4 ⊢ (Ⅎ𝑦𝜑 → ([𝑥 / 𝑦]𝜑 ↔ 𝜑)) | |
2 | 1 | alimi 1806 | . . 3 ⊢ (∀𝑥Ⅎ𝑦𝜑 → ∀𝑥([𝑥 / 𝑦]𝜑 ↔ 𝜑)) |
3 | albi 1813 | . . 3 ⊢ (∀𝑥([𝑥 / 𝑦]𝜑 ↔ 𝜑) → (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑥𝜑)) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑥𝜑)) |
5 | wl-sb9v 37490 | . 2 ⊢ (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑) | |
6 | 4, 5 | bitr3di 286 | 1 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∀wal 1533 Ⅎwnf 1778 [wsb 2060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-11 2153 ax-12 2173 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1775 df-nf 1779 df-sb 2061 |
This theorem is referenced by: wl-sb8eft 37492 |
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