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Metamath Proof Explorer

Theorem syl2an2

Description: syl2an with antecedents in standard conjunction form. (Contributed by Alan Sare, 27-Aug-2016)

Ref Expression
Hypotheses syl2an2.1 ⊢ ( 𝜑 → 𝜓 )
syl2an2.2 ⊢ ( ( 𝜒 ∧ 𝜑 ) → 𝜃 )
syl2an2.3 ⊢ ( ( 𝜓 ∧ 𝜃 ) → 𝜏 )
Assertion syl2an2 ( ( 𝜒 ∧ 𝜑 ) → 𝜏 )

Proof

Step Hyp Ref Expression
1 syl2an2.1 ⊢ ( 𝜑 → 𝜓 )
2 syl2an2.2 ⊢ ( ( 𝜒 ∧ 𝜑 ) → 𝜃 )
3 syl2an2.3 ⊢ ( ( 𝜓 ∧ 𝜃 ) → 𝜏 )
4 1 adantl ⊢ ( ( 𝜒 ∧ 𝜑 ) → 𝜓 )
5 4 2 3 syl2anc ⊢ ( ( 𝜒 ∧ 𝜑 ) → 𝜏 )