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Mirrors > Home > ILE Home > Th. List > simpl2im | 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 110 | . 2 ⊢ ((𝜓 ∧ 𝜒) → 𝜒) | |
3 | simpl2im.2 | . 2 ⊢ (𝜒 → 𝜃) | |
4 | 1, 2, 3 | 3syl 17 | 1 ⊢ (𝜑 → 𝜃) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia2 107 |
This theorem is referenced by: ctssdccl 7107 enumct 7111 djuen 7207 ndvdssub 11927 sgrpidmndm 12753 xmeteq0 13730 xmettri2 13732 metcnpi 13886 metcnpi2 13887 dvbssntrcntop 14024 |
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