Metamath Proof Explorer


Theorem syl2anr

Description: A double syllogism inference. For an implication-only version, see syl2imc . (Contributed by NM, 17-Sep-2013)

Ref Expression
Hypotheses syl2an.1 ( 𝜑𝜓 )
syl2an.2 ( 𝜏𝜒 )
syl2an.3 ( ( 𝜓𝜒 ) → 𝜃 )
Assertion syl2anr ( ( 𝜏𝜑 ) → 𝜃 )

Proof

Step Hyp Ref Expression
1 syl2an.1 ( 𝜑𝜓 )
2 syl2an.2 ( 𝜏𝜒 )
3 syl2an.3 ( ( 𝜓𝜒 ) → 𝜃 )
4 1 2 3 syl2an ( ( 𝜑𝜏 ) → 𝜃 )
5 4 ancoms ( ( 𝜏𝜑 ) → 𝜃 )