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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 2500 with a disjoint variable condition, not requiring ax-13 2333. (Contributed 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 2253 | . 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | |
3 | sbequ12 2228 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
4 | 1, 2, 3 | cbvalv1 2311 | 1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∀wal 1599 Ⅎwnf 1827 [wsb 2011 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-10 2134 ax-11 2149 ax-12 2162 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-ex 1824 df-nf 1828 df-sb 2012 |
This theorem is referenced by: sbnf2 2325 sb8eulem 2632 |
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