Metamath Proof Explorer


Theorem 3imtr3g

Description: More general version of 3imtr3i . Useful for converting definitions in a formula. (Contributed by NM, 20-May-1996) (Proof shortened by Wolf Lammen, 20-Dec-2013)

Ref Expression
Hypotheses 3imtr3g.1 ( 𝜑 → ( 𝜓𝜒 ) )
3imtr3g.2 ( 𝜓𝜃 )
3imtr3g.3 ( 𝜒𝜏 )
Assertion 3imtr3g ( 𝜑 → ( 𝜃𝜏 ) )

Proof

Step Hyp Ref Expression
1 3imtr3g.1 ( 𝜑 → ( 𝜓𝜒 ) )
2 3imtr3g.2 ( 𝜓𝜃 )
3 3imtr3g.3 ( 𝜒𝜏 )
4 2 1 syl5bir ( 𝜑 → ( 𝜃𝜒 ) )
5 4 3 syl6ib ( 𝜑 → ( 𝜃𝜏 ) )