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Mirrors > Home > MPE Home > Th. List > ibibr | Structured version Visualization version GIF version |
Description: Implication in terms of implication and biconditional. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Wolf Lammen, 21-Dec-2013.) |
Ref | Expression |
---|---|
ibibr | ⊢ ((𝜑 → 𝜓) ↔ (𝜑 → (𝜓 ↔ 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm5.501 367 | . . 3 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ↔ 𝜓))) | |
2 | bicom 221 | . . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ (𝜓 ↔ 𝜑)) | |
3 | 1, 2 | bitrdi 287 | . 2 ⊢ (𝜑 → (𝜓 ↔ (𝜓 ↔ 𝜑))) |
4 | 3 | pm5.74i 270 | 1 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 → (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 |
This theorem depends on definitions: df-bi 206 |
This theorem is referenced by: tbt 370 rabxfrd 5344 ufileu 23068 abnotbtaxb 44378 |
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