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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 to apply Frege-style definitions and explore their consequences. A closed form of biimpri 227. 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 42594 | . 2 ⊢ ((𝜓 ↔ 𝜑) → (𝜓 → 𝜑)) | |
2 | frege56aid 42606 | . 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 205 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-frege1 42526 ax-frege2 42527 ax-frege8 42545 ax-frege52a 42593 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-ifp 1062 df-tru 1544 df-fal 1554 |
This theorem is referenced by: frege68a 42622 frege68b 42649 frege68c 42667 frege100 42699 |
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