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Mirrors > Home > MPE Home > Th. List > sbtr | Structured version Visualization version GIF version |
Description: A partial converse to sbt 2552. If the substitution of a variable for a non-free one in a wff gives a theorem, then the original wff is a theorem. (Contributed by BJ, 15-Sep-2018.) |
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 2553 | . 2 ⊢ (∀𝑦[𝑦 / 𝑥]𝜑 → 𝜑) |
3 | sbtr.1 | . 2 ⊢ [𝑦 / 𝑥]𝜑 | |
4 | 2, 3 | mpg 1898 | 1 ⊢ 𝜑 |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnf 1884 [wsb 2069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-10 2194 ax-12 2222 ax-13 2391 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-ex 1881 df-nf 1885 df-sb 2070 |
This theorem is referenced by: (None) |
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