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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 229. 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 40082 | . 2 ⊢ ((𝜓 ↔ 𝜑) → (𝜓 → 𝜑)) | |
2 | frege56aid 40094 | . 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 207 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-frege1 40014 ax-frege2 40015 ax-frege8 40033 ax-frege52a 40081 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ifp 1055 df-tru 1531 df-fal 1541 |
This theorem is referenced by: frege68a 40110 frege68b 40137 frege68c 40155 frege100 40187 |
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