Metamath Proof Explorer

Theorem stoic4b

Description: Stoic logic Thema 4 version b. This is version b, which is with the phrase "or both". See stoic4a for more information. (Contributed by David A. Wheeler, 17-Feb-2019)

	Ref	Expression
Hypotheses	stoic4b.1	$⊢ (φ \land ψ) \to χ$
	stoic4b.2	$⊢ ((χ \land φ \land ψ) \land θ) \to τ$
Assertion	stoic4b	$⊢ (φ \land ψ \land θ) \to τ$

Proof

Step	Hyp	Ref	Expression
1		stoic4b.1	$⊢ (φ \land ψ) \to χ$
2		stoic4b.2	$⊢ ((χ \land φ \land ψ) \land θ) \to τ$
3	1	3adant3	$⊢ (φ \land ψ \land θ) \to χ$
4		simp1	$⊢ (φ \land ψ \land θ) \to φ$
5		simp2	$⊢ (φ \land ψ \land θ) \to ψ$
6		simp3	$⊢ (φ \land ψ \land θ) \to θ$
7	3 4 5 6 2	syl31anc	$⊢ (φ \land ψ \land θ) \to τ$