Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > frege57aid | Structured version Visualization version GIF version |
Description: This is the all imporant formula which allows us to apply Frege-style definitions and explore their consequences. A closed form of biimpri 217. Proposition 57 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege57aid | ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege52aid 37172 | . 2 ⊢ ((𝜓 ↔ 𝜑) → (𝜓 → 𝜑)) | |
2 | frege56aid 37184 | . 2 ⊢ (((𝜓 ↔ 𝜑) → (𝜓 → 𝜑)) → ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-frege1 37104 ax-frege2 37105 ax-frege8 37123 ax-frege52a 37171 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-ifp 1007 df-tru 1478 df-fal 1481 |
This theorem is referenced by: frege68a 37200 frege68b 37227 frege68c 37245 frege100 37277 |
Copyright terms: Public domain | W3C validator |