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Mirrors > Home > ILE Home > Th. List > ibib | GIF version |
Description: Implication in terms of implication and biconditional. (Contributed by NM, 31-Mar-1994.) (Proof shortened by Wolf Lammen, 24-Jan-2013.) |
Ref | Expression |
---|---|
ibib | ⊢ ((𝜑 → 𝜓) ↔ (𝜑 → (𝜑 ↔ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm5.501 233 | . 2 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ↔ 𝜓))) | |
2 | 1 | pm5.74i 169 | 1 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 → (𝜑 ↔ 𝜓))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: (None) |
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