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Mirrors > Home > MPE Home > Th. List > pm5.75OLD | Structured version Visualization version GIF version |
Description: Obsolete proof of pm5.75 974 as of 12-Feb-2021. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 23-Dec-2012.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
pm5.75OLD | ⊢ (((𝜒 → ¬ 𝜓) ∧ (𝜑 ↔ (𝜓 ∨ 𝜒))) → ((𝜑 ∧ ¬ 𝜓) ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | anbi1 739 | . . 3 ⊢ ((𝜑 ↔ (𝜓 ∨ 𝜒)) → ((𝜑 ∧ ¬ 𝜓) ↔ ((𝜓 ∨ 𝜒) ∧ ¬ 𝜓))) | |
2 | orcom 401 | . . . . 5 ⊢ ((𝜓 ∨ 𝜒) ↔ (𝜒 ∨ 𝜓)) | |
3 | 2 | anbi1i 727 | . . . 4 ⊢ (((𝜓 ∨ 𝜒) ∧ ¬ 𝜓) ↔ ((𝜒 ∨ 𝜓) ∧ ¬ 𝜓)) |
4 | pm5.61 745 | . . . 4 ⊢ (((𝜒 ∨ 𝜓) ∧ ¬ 𝜓) ↔ (𝜒 ∧ ¬ 𝜓)) | |
5 | 3, 4 | bitri 263 | . . 3 ⊢ (((𝜓 ∨ 𝜒) ∧ ¬ 𝜓) ↔ (𝜒 ∧ ¬ 𝜓)) |
6 | 1, 5 | syl6bb 275 | . 2 ⊢ ((𝜑 ↔ (𝜓 ∨ 𝜒)) → ((𝜑 ∧ ¬ 𝜓) ↔ (𝜒 ∧ ¬ 𝜓))) |
7 | pm4.71 660 | . . . 4 ⊢ ((𝜒 → ¬ 𝜓) ↔ (𝜒 ↔ (𝜒 ∧ ¬ 𝜓))) | |
8 | 7 | biimpi 205 | . . 3 ⊢ ((𝜒 → ¬ 𝜓) → (𝜒 ↔ (𝜒 ∧ ¬ 𝜓))) |
9 | 8 | bicomd 212 | . 2 ⊢ ((𝜒 → ¬ 𝜓) → ((𝜒 ∧ ¬ 𝜓) ↔ 𝜒)) |
10 | 6, 9 | sylan9bbr 733 | 1 ⊢ (((𝜒 → ¬ 𝜓) ∧ (𝜑 ↔ (𝜓 ∨ 𝜒))) → ((𝜑 ∧ ¬ 𝜓) ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 |
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 196 df-or 384 df-an 385 |
This theorem is referenced by: (None) |
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