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Metamath Proof Explorer

Theorem syl2an2r

Description: syl2anr with antecedents in standard conjunction form. (Contributed by Alan Sare, 27-Aug-2016) (Proof shortened by Wolf Lammen, 28-Mar-2022)

	Ref	Expression
Hypotheses	syl2an2r.1	$⊢ φ \to ψ$
	syl2an2r.2	$⊢ (φ \land χ) \to θ$
	syl2an2r.3	$⊢ (ψ \land θ) \to τ$
Assertion	syl2an2r	$⊢ (φ \land χ) \to τ$

Proof

Step	Hyp	Ref	Expression
1		syl2an2r.1	$⊢ φ \to ψ$
2		syl2an2r.2	$⊢ (φ \land χ) \to θ$
3		syl2an2r.3	$⊢ (ψ \land θ) \to τ$
4	1 3	sylan	$⊢ (φ \land θ) \to τ$
5	2 4	syldan	$⊢ (φ \land χ) \to τ$