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Mirrors > Home > MPE Home > Th. List > sbtr | Structured version Visualization version GIF version |
Description: A partial converse to sbt 2062. If the substitution of a variable for a nonfree one in a wff gives a theorem, then the original wff is a theorem. Usage of this theorem is discouraged because it depends on ax-13 2367. (Contributed by BJ, 15-Sep-2018.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sbtr.nf | ⊢ Ⅎ𝑦𝜑 |
sbtr.1 | ⊢ [𝑦 / 𝑥]𝜑 |
Ref | Expression |
---|---|
sbtr | ⊢ 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbtr.nf | . . 3 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | sbtrt 2510 | . 2 ⊢ (∀𝑦[𝑦 / 𝑥]𝜑 → 𝜑) |
3 | sbtr.1 | . 2 ⊢ [𝑦 / 𝑥]𝜑 | |
4 | 2, 3 | mpg 1792 | 1 ⊢ 𝜑 |
Colors of variables: wff setvar class |
Syntax hints: 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 1964 ax-7 2004 ax-10 2130 ax-12 2167 ax-13 2367 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-ex 1775 df-nf 1779 df-sb 2061 |
This theorem is referenced by: (None) |
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