Metamath Proof Explorer


Theorem ad5ant125

Description: Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017) (Proof shortened by Wolf Lammen, 23-Jun-2022)

Ref Expression
Hypothesis ad5ant.1 ( ( 𝜑𝜓𝜒 ) → 𝜃 )
Assertion ad5ant125 ( ( ( ( ( 𝜑𝜓 ) ∧ 𝜏 ) ∧ 𝜂 ) ∧ 𝜒 ) → 𝜃 )

Proof

Step Hyp Ref Expression
1 ad5ant.1 ( ( 𝜑𝜓𝜒 ) → 𝜃 )
2 1 3expia ( ( 𝜑𝜓 ) → ( 𝜒𝜃 ) )
3 2 2a1d ( ( 𝜑𝜓 ) → ( 𝜏 → ( 𝜂 → ( 𝜒𝜃 ) ) ) )
4 3 imp41 ( ( ( ( ( 𝜑𝜓 ) ∧ 𝜏 ) ∧ 𝜂 ) ∧ 𝜒 ) → 𝜃 )