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Metamath Proof Explorer

Theorem 3jcad

Description: Deduction conjoining the consequents of three implications. (Contributed by NM, 25-Sep-2005)

Ref Expression
Hypotheses 3jcad.1 ⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
3jcad.2 ⊢ ( 𝜑 → ( 𝜓 → 𝜃 ) )
3jcad.3 ⊢ ( 𝜑 → ( 𝜓 → 𝜏 ) )
Assertion 3jcad ( 𝜑 → ( 𝜓 → ( 𝜒 ∧ 𝜃 ∧ 𝜏 ) ) )

Proof

Step Hyp Ref Expression
1 3jcad.1 ⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2 3jcad.2 ⊢ ( 𝜑 → ( 𝜓 → 𝜃 ) )
3 3jcad.3 ⊢ ( 𝜑 → ( 𝜓 → 𝜏 ) )
4 1 imp ⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
5 2 imp ⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜃 )
6 3 imp ⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜏 )
7 4 5 6 3jca ⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜒 ∧ 𝜃 ∧ 𝜏 ) )
8 7 ex ⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 ∧ 𝜃 ∧ 𝜏 ) ) )