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Mirrors > Home > ILE Home > Th. List > dedlemb | GIF version |
Description: Lemma for iffalse 3339. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 7-May-2011.) |
Ref | Expression |
---|---|
dedlemb | ⊢ (¬ 𝜑 → (𝜒 ↔ ((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | olc 632 | . . 3 ⊢ ((𝜒 ∧ ¬ 𝜑) → ((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑))) | |
2 | 1 | expcom 109 | . 2 ⊢ (¬ 𝜑 → (𝜒 → ((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑)))) |
3 | pm2.21 547 | . . . 4 ⊢ (¬ 𝜑 → (𝜑 → 𝜒)) | |
4 | 3 | adantld 263 | . . 3 ⊢ (¬ 𝜑 → ((𝜓 ∧ 𝜑) → 𝜒)) |
5 | simpl 102 | . . . 4 ⊢ ((𝜒 ∧ ¬ 𝜑) → 𝜒) | |
6 | 5 | a1i 9 | . . 3 ⊢ (¬ 𝜑 → ((𝜒 ∧ ¬ 𝜑) → 𝜒)) |
7 | 4, 6 | jaod 637 | . 2 ⊢ (¬ 𝜑 → (((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑)) → 𝜒)) |
8 | 2, 7 | impbid 120 | 1 ⊢ (¬ 𝜑 → (𝜒 ↔ ((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑)))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 ↔ wb 98 ∨ wo 629 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in2 545 ax-io 630 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: iffalse 3339 |
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