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Metamath Proof Explorer

Theorem jca

Description: Deduce conjunction of the consequents of two implications ("join consequents with 'and'"). Deduction form of pm3.2 and pm3.2i . Its associated deduction is jcad . Equivalent to the natural deduction rule /\ I ( /\ introduction), see natded . (Contributed by NM, 3-Jan-1993) (Proof shortened by Wolf Lammen, 25-Oct-2012)

	Ref	Expression
Hypotheses	jca.1	⊢ ( 𝜑 → 𝜓 )
	jca.2	⊢ ( 𝜑 → 𝜒 )
Assertion	jca	⊢ ( 𝜑 → ( 𝜓 ∧ 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		jca.1	⊢ ( 𝜑 → 𝜓 )
2		jca.2	⊢ ( 𝜑 → 𝜒 )
3		pm3.2	⊢ ( 𝜓 → ( 𝜒 → ( 𝜓 ∧ 𝜒 ) ) )
4	1 2 3	sylc	⊢ ( 𝜑 → ( 𝜓 ∧ 𝜒 ) )