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Mirrors > Home > ILE Home > Th. List > dedlem0a | GIF version |
Description: Alternate version of dedlema 969. (Contributed by NM, 2-Apr-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 4-Dec-2012.) |
Ref | Expression |
---|---|
dedlem0a | ⊢ (𝜑 → (𝜓 ↔ ((𝜒 → 𝜑) → (𝜓 ∧ 𝜑)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iba 300 | . 2 ⊢ (𝜑 → (𝜓 ↔ (𝜓 ∧ 𝜑))) | |
2 | ax-1 6 | . . 3 ⊢ (𝜑 → (𝜒 → 𝜑)) | |
3 | biimt 241 | . . 3 ⊢ ((𝜒 → 𝜑) → ((𝜓 ∧ 𝜑) ↔ ((𝜒 → 𝜑) → (𝜓 ∧ 𝜑)))) | |
4 | 2, 3 | syl 14 | . 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜑) ↔ ((𝜒 → 𝜑) → (𝜓 ∧ 𝜑)))) |
5 | 1, 4 | bitrd 188 | 1 ⊢ (𝜑 → (𝜓 ↔ ((𝜒 → 𝜑) → (𝜓 ∧ 𝜑)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: (None) |
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