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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege55cor1a | Structured version Visualization version GIF version |
Description: Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege55cor1a | ⊢ ((𝜑 ↔ 𝜓) → (𝜓 ↔ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege55a 42228 | . 2 ⊢ ((𝜑 ↔ 𝜓) → if-(𝜓, 𝜑, ¬ 𝜑)) | |
2 | frege55lem1a 42226 | . 2 ⊢ (((𝜑 ↔ 𝜓) → if-(𝜓, 𝜑, ¬ 𝜑)) → ((𝜑 ↔ 𝜓) → (𝜓 ↔ 𝜑))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 ↔ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 if-wif 1062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-frege8 42169 ax-frege28 42190 ax-frege52a 42217 ax-frege54a 42222 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ifp 1063 |
This theorem is referenced by: frege56a 42231 |
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