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Mathbox for Wolf Lammen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-sbnf1 | Structured version Visualization version GIF version |
Description: Two ways expressing that 𝑥 is effectively not free in 𝜑. Simplified version of sbnf2 2354. Note: This theorem shows that sbnf2 2354 has unnecessary distinct variable constraints. (Contributed by Wolf Lammen, 28-Jul-2019.) |
Ref | Expression |
---|---|
wl-sbnf1 | ⊢ (∀𝑥Ⅎ𝑦𝜑 → (Ⅎ𝑥𝜑 ↔ ∀𝑥∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nf5 2278 | . 2 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑)) | |
2 | nfa1 2148 | . . 3 ⊢ Ⅎ𝑥∀𝑥Ⅎ𝑦𝜑 | |
3 | wl-sbhbt 36116 | . . 3 ⊢ (∀𝑥Ⅎ𝑦𝜑 → ((𝜑 → ∀𝑥𝜑) ↔ ∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑))) | |
4 | 2, 3 | albid 2215 | . 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑))) |
5 | 1, 4 | bitrid 282 | 1 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (Ⅎ𝑥𝜑 ↔ ∀𝑥∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1539 Ⅎwnf 1785 [wsb 2067 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-10 2137 ax-11 2154 ax-12 2171 ax-13 2370 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-ex 1782 df-nf 1786 df-sb 2068 |
This theorem is referenced by: (None) |
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