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Mirrors > Home > ILE Home > Th. List > dedlema | GIF version |
Description: Lemma for iftrue 3356. (Contributed by NM, 26-Jun-2002.) (Proof shortened by Andrew Salmon, 7-May-2011.) |
Ref | Expression |
---|---|
dedlema | ⊢ (𝜑 → (𝜓 ↔ ((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orc 665 | . . 3 ⊢ ((𝜓 ∧ 𝜑) → ((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑))) | |
2 | 1 | expcom 114 | . 2 ⊢ (𝜑 → (𝜓 → ((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑)))) |
3 | simpl 107 | . . . 4 ⊢ ((𝜓 ∧ 𝜑) → 𝜓) | |
4 | 3 | a1i 9 | . . 3 ⊢ (𝜑 → ((𝜓 ∧ 𝜑) → 𝜓)) |
5 | pm2.24 583 | . . . 4 ⊢ (𝜑 → (¬ 𝜑 → 𝜓)) | |
6 | 5 | adantld 272 | . . 3 ⊢ (𝜑 → ((𝜒 ∧ ¬ 𝜑) → 𝜓)) |
7 | 4, 6 | jaod 669 | . 2 ⊢ (𝜑 → (((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑)) → 𝜓)) |
8 | 2, 7 | impbid 127 | 1 ⊢ (𝜑 → (𝜓 ↔ ((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑)))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 ↔ wb 103 ∨ wo 661 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in2 577 ax-io 662 |
This theorem depends on definitions: df-bi 115 |
This theorem is referenced by: iftrue 3356 |
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