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Mirrors > Home > ILE Home > Th. List > anim12ii | GIF version |
Description: Conjoin antecedents and consequents in a deduction. (Contributed by NM, 11-Nov-2007.) (Proof shortened by Wolf Lammen, 19-Jul-2013.) |
Ref | Expression |
---|---|
anim12ii.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
anim12ii.2 | ⊢ (𝜃 → (𝜓 → 𝜏)) |
Ref | Expression |
---|---|
anim12ii | ⊢ ((𝜑 ∧ 𝜃) → (𝜓 → (𝜒 ∧ 𝜏))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | anim12ii.1 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
2 | 1 | adantr 274 | . 2 ⊢ ((𝜑 ∧ 𝜃) → (𝜓 → 𝜒)) |
3 | anim12ii.2 | . . 3 ⊢ (𝜃 → (𝜓 → 𝜏)) | |
4 | 3 | adantl 275 | . 2 ⊢ ((𝜑 ∧ 𝜃) → (𝜓 → 𝜏)) |
5 | 2, 4 | jcad 305 | 1 ⊢ ((𝜑 ∧ 𝜃) → (𝜓 → (𝜒 ∧ 𝜏))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 |
This theorem is referenced by: euim 2087 elex22 2745 tz7.2 4337 funcnvuni 5265 bj-findis 13974 |
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