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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 → 𝜓 )
	syl2an2r.2	⊢ ( ( 𝜑 ∧ 𝜒 ) → 𝜃 )
	syl2an2r.3	⊢ ( ( 𝜓 ∧ 𝜃 ) → 𝜏 )
Assertion	syl2an2r	⊢ ( ( 𝜑 ∧ 𝜒 ) → 𝜏 )

Proof

Step	Hyp	Ref	Expression
1		syl2an2r.1	⊢ ( 𝜑 → 𝜓 )
2		syl2an2r.2	⊢ ( ( 𝜑 ∧ 𝜒 ) → 𝜃 )
3		syl2an2r.3	⊢ ( ( 𝜓 ∧ 𝜃 ) → 𝜏 )
4	1 3	sylan	⊢ ( ( 𝜑 ∧ 𝜃 ) → 𝜏 )
5	2 4	syldan	⊢ ( ( 𝜑 ∧ 𝜒 ) → 𝜏 )