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Mathbox for Wolf Lammen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-sb8t | Structured version Visualization version GIF version |
Description: Substitution of variable in universal quantifier. Closed form of sb8 2520. (Contributed by Wolf Lammen, 27-Jul-2019.) |
Ref | Expression |
---|---|
wl-sb8t | ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfa1 2149 | . 2 ⊢ Ⅎ𝑥∀𝑥Ⅎ𝑦𝜑 | |
2 | nfnf1 2152 | . . 3 ⊢ Ⅎ𝑦Ⅎ𝑦𝜑 | |
3 | 2 | nfal 2322 | . 2 ⊢ Ⅎ𝑦∀𝑥Ⅎ𝑦𝜑 |
4 | sp 2181 | . 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦𝜑) | |
5 | wl-nfs1t 37518 | . . 3 ⊢ (Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑) | |
6 | 5 | sps 2183 | . 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑) |
7 | sbequ12 2249 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
8 | 7 | a1i 11 | . 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))) |
9 | 1, 3, 4, 6, 8 | cbv2 2406 | 1 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∀wal 1535 Ⅎwnf 1780 [wsb 2062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-10 2139 ax-11 2155 ax-12 2175 ax-13 2375 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1777 df-nf 1781 df-sb 2063 |
This theorem is referenced by: wl-sb8et 37534 wl-sbhbt 37535 |
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