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Mirrors > Home > MPE Home > Th. List > elimhOLD | Structured version Visualization version GIF version |
Description: Old version of elimh 1024. Obsolete as of 16-Mar-2021. (Contributed by NM, 26-Jun-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
elimhOLD.1 | ⊢ ((𝜑 ↔ ((𝜑 ∧ 𝜒) ∨ (𝜓 ∧ ¬ 𝜒))) → (𝜒 ↔ 𝜏)) |
elimhOLD.2 | ⊢ ((𝜓 ↔ ((𝜑 ∧ 𝜒) ∨ (𝜓 ∧ ¬ 𝜒))) → (𝜃 ↔ 𝜏)) |
elimhOLD.3 | ⊢ 𝜃 |
Ref | Expression |
---|---|
elimhOLD | ⊢ 𝜏 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dedlema 993 | . . . 4 ⊢ (𝜒 → (𝜑 ↔ ((𝜑 ∧ 𝜒) ∨ (𝜓 ∧ ¬ 𝜒)))) | |
2 | elimhOLD.1 | . . . 4 ⊢ ((𝜑 ↔ ((𝜑 ∧ 𝜒) ∨ (𝜓 ∧ ¬ 𝜒))) → (𝜒 ↔ 𝜏)) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜒 → (𝜒 ↔ 𝜏)) |
4 | 3 | ibi 255 | . 2 ⊢ (𝜒 → 𝜏) |
5 | elimhOLD.3 | . . 3 ⊢ 𝜃 | |
6 | dedlemb 994 | . . . 4 ⊢ (¬ 𝜒 → (𝜓 ↔ ((𝜑 ∧ 𝜒) ∨ (𝜓 ∧ ¬ 𝜒)))) | |
7 | elimhOLD.2 | . . . 4 ⊢ ((𝜓 ↔ ((𝜑 ∧ 𝜒) ∨ (𝜓 ∧ ¬ 𝜒))) → (𝜃 ↔ 𝜏)) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ (¬ 𝜒 → (𝜃 ↔ 𝜏)) |
9 | 5, 8 | mpbii 222 | . 2 ⊢ (¬ 𝜒 → 𝜏) |
10 | 4, 9 | pm2.61i 175 | 1 ⊢ 𝜏 |
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: con3OLD 1029 |
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