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Mirrors > Home > MPE Home > Th. List > dedlemb | Structured version Visualization version GIF version |
Description: Lemma for weak deduction theorem. See also ifpfal 1074. (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 865 | . . 3 ⊢ ((𝜒 ∧ ¬ 𝜑) → ((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑))) | |
2 | 1 | expcom 414 | . 2 ⊢ (¬ 𝜑 → (𝜒 → ((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑)))) |
3 | pm2.21 123 | . . . 4 ⊢ (¬ 𝜑 → (𝜑 → 𝜒)) | |
4 | 3 | adantld 491 | . . 3 ⊢ (¬ 𝜑 → ((𝜓 ∧ 𝜑) → 𝜒)) |
5 | simpl 483 | . . . 4 ⊢ ((𝜒 ∧ ¬ 𝜑) → 𝜒) | |
6 | 5 | a1i 11 | . . 3 ⊢ (¬ 𝜑 → ((𝜒 ∧ ¬ 𝜑) → 𝜒)) |
7 | 4, 6 | jaod 856 | . 2 ⊢ (¬ 𝜑 → (((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑)) → 𝜒)) |
8 | 2, 7 | impbid 211 | 1 ⊢ (¬ 𝜑 → (𝜒 ↔ ((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑)))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 844 |
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 df-an 397 df-or 845 |
This theorem is referenced by: cases2 1045 pm4.42 1051 iffalse 4468 |
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