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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	$⊢ φ \to ψ$
	jca.2	$⊢ φ \to χ$
Assertion	jca	$⊢ φ \to (ψ \land χ)$

Proof

Step	Hyp	Ref	Expression
1		jca.1	$⊢ φ \to ψ$
2		jca.2	$⊢ φ \to χ$
3		pm3.2	$⊢ ψ \to (χ \to (ψ \land χ))$
4	1 2 3	sylc	$⊢ φ \to (ψ \land χ)$