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Mirrors > Home > MPE Home > Th. List > simpl2im | Structured version Visualization version GIF version |
Description: Implication from an eliminated conjunct implied by the antecedent. (Contributed by BJ/AV, 5-Apr-2021.) |
Ref | Expression |
---|---|
simpl2im.1 | ⊢ (𝜑 → (𝜓 ∧ 𝜒)) |
simpl2im.2 | ⊢ (𝜒 → 𝜃) |
Ref | Expression |
---|---|
simpl2im | ⊢ (𝜑 → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl2im.1 | . 2 ⊢ (𝜑 → (𝜓 ∧ 𝜒)) | |
2 | simpr 476 | . 2 ⊢ ((𝜓 ∧ 𝜒) → 𝜒) | |
3 | simpl2im.2 | . 2 ⊢ (𝜒 → 𝜃) | |
4 | 1, 2, 3 | 3syl 18 | 1 ⊢ (𝜑 → 𝜃) |
Colors of variables: wff setvar class |
Syntax hints: → 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: dvdsaddre2b 14867 nbgrcl 40559 usgr2trlncrct 41009 wwlksnextproplem3 41117 erclwwlksnsym 41254 erclwwlksntr 41255 av-numclwlk2lem2f 41533 |
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