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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege52aid | Structured version Visualization version GIF version |
Description: The case when the content of 𝜑 is identical with the content of 𝜓 and in which 𝜑 is affirmed and 𝜓 is denied does not take place. Identical to biimp 204. Part of Axiom 52 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege52aid | ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-frege52a 37171 | . 2 ⊢ ((𝜑 ↔ 𝜓) → (if-(𝜑, ⊤, ⊥) → if-(𝜓, ⊤, ⊥))) | |
2 | ifpid2 36834 | . 2 ⊢ (𝜑 ↔ if-(𝜑, ⊤, ⊥)) | |
3 | ifpid2 36834 | . 2 ⊢ (𝜓 ↔ if-(𝜓, ⊤, ⊥)) | |
4 | 1, 2, 3 | 3imtr4g 284 | 1 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 if-wif 1006 ⊤wtru 1476 ⊥wfal 1480 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 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: frege53aid 37173 frege57aid 37186 frege75 37252 frege89 37266 frege105 37282 |
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