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Mirrors > Home > MPE Home > Th. List > sb8v | Structured version Visualization version GIF version |
Description: Substitution of variable in universal quantifier. Version of sb8 2555 with a disjoint variable condition, not requiring ax-13 2386. (Contributed by NM, 16-May-1993.) (Revised by Wolf Lammen, 19-Jan-2023.) |
Ref | Expression |
---|---|
sb8v.nf | ⊢ Ⅎ𝑦𝜑 |
Ref | Expression |
---|---|
sb8v | ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb8v.nf | . 2 ⊢ Ⅎ𝑦𝜑 | |
2 | nfs1v 2156 | . 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | |
3 | sbequ12 2249 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
4 | 1, 2, 3 | cbvalv1 2357 | 1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∀wal 1531 Ⅎwnf 1780 [wsb 2065 |
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 1907 ax-6 1966 ax-7 2011 ax-10 2141 ax-11 2157 ax-12 2173 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1777 df-nf 1781 df-sb 2066 |
This theorem is referenced by: sbnf2 2373 sb8eulem 2680 abv 3504 mo5f 30252 ax11-pm2 34159 bj-nfcf 34242 sbcalf 35391 |
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