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Mirrors > Home > MPE Home > Th. List > ex-natded5.13-2 | Structured version Visualization version GIF version |
Description: A more efficient proof of Theorem 5.13 of [Clemente] p. 20. Compare with ex-natded5.13 26664. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ex-natded5.13.1 | ⊢ (𝜑 → (𝜓 ∨ 𝜒)) |
ex-natded5.13.2 | ⊢ (𝜑 → (𝜓 → 𝜃)) |
ex-natded5.13.3 | ⊢ (𝜑 → (¬ 𝜏 → ¬ 𝜒)) |
Ref | Expression |
---|---|
ex-natded5.13-2 | ⊢ (𝜑 → (𝜃 ∨ 𝜏)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ex-natded5.13.1 | . 2 ⊢ (𝜑 → (𝜓 ∨ 𝜒)) | |
2 | ex-natded5.13.2 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜃)) | |
3 | ex-natded5.13.3 | . . . 4 ⊢ (𝜑 → (¬ 𝜏 → ¬ 𝜒)) | |
4 | 3 | con4d 113 | . . 3 ⊢ (𝜑 → (𝜒 → 𝜏)) |
5 | 2, 4 | orim12d 879 | . 2 ⊢ (𝜑 → ((𝜓 ∨ 𝜒) → (𝜃 ∨ 𝜏))) |
6 | 1, 5 | mpd 15 | 1 ⊢ (𝜑 → (𝜃 ∨ 𝜏)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 382 |
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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