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Mirrors > Home > MPE Home > Th. List > sbtr | Structured version Visualization version GIF version |
Description: A partial converse to sbt 2067. If the substitution of a variable for a non-free 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 2386. (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 2553 | . 2 ⊢ (∀𝑦[𝑦 / 𝑥]𝜑 → 𝜑) |
3 | sbtr.1 | . 2 ⊢ [𝑦 / 𝑥]𝜑 | |
4 | 2, 3 | mpg 1794 | 1 ⊢ 𝜑 |
Colors of variables: wff setvar class |
Syntax hints: 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-12 2172 ax-13 2386 |
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: (None) |
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