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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 1076. (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 867 | . . 3 ⊢ ((𝜒 ∧ ¬ 𝜑) → ((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑))) | |
2 | 1 | expcom 417 | . 2 ⊢ (¬ 𝜑 → (𝜒 → ((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑)))) |
3 | pm2.21 123 | . . . 4 ⊢ (¬ 𝜑 → (𝜑 → 𝜒)) | |
4 | 3 | adantld 494 | . . 3 ⊢ (¬ 𝜑 → ((𝜓 ∧ 𝜑) → 𝜒)) |
5 | simpl 486 | . . . 4 ⊢ ((𝜒 ∧ ¬ 𝜑) → 𝜒) | |
6 | 5 | a1i 11 | . . 3 ⊢ (¬ 𝜑 → ((𝜒 ∧ ¬ 𝜑) → 𝜒)) |
7 | 4, 6 | jaod 858 | . 2 ⊢ (¬ 𝜑 → (((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑)) → 𝜒)) |
8 | 2, 7 | impbid 215 | 1 ⊢ (¬ 𝜑 → (𝜒 ↔ ((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑)))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 846 |
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 210 df-an 400 df-or 847 |
This theorem is referenced by: cases2 1047 pm4.42 1053 iffalse 4424 |
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