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Mirrors > Home > MPE Home > Th. List > ex-natded5.8-2 | Structured version Visualization version GIF version |
Description: A more efficient proof of Theorem 5.8 of [Clemente] p. 20. For a longer line-by-line translation, see ex-natded5.8 26662. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ex-natded5.8.1 | ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → ¬ 𝜃)) |
ex-natded5.8.2 | ⊢ (𝜑 → (𝜏 → 𝜃)) |
ex-natded5.8.3 | ⊢ (𝜑 → 𝜒) |
ex-natded5.8.4 | ⊢ (𝜑 → 𝜏) |
Ref | Expression |
---|---|
ex-natded5.8-2 | ⊢ (𝜑 → ¬ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ex-natded5.8.4 | . . 3 ⊢ (𝜑 → 𝜏) | |
2 | ex-natded5.8.2 | . . 3 ⊢ (𝜑 → (𝜏 → 𝜃)) | |
3 | 1, 2 | mpd 15 | . 2 ⊢ (𝜑 → 𝜃) |
4 | ex-natded5.8.3 | . . 3 ⊢ (𝜑 → 𝜒) | |
5 | ex-natded5.8.1 | . . 3 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → ¬ 𝜃)) | |
6 | 4, 5 | mpan2d 706 | . 2 ⊢ (𝜑 → (𝜓 → ¬ 𝜃)) |
7 | 3, 6 | mt2d 130 | 1 ⊢ (𝜑 → ¬ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ 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-an 385 |
This theorem is referenced by: (None) |
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