Theorem ad5ant125 1364

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

Hypothesis

Ref	Expression
ad5ant.1	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Assertion

Ref	Expression
ad5ant125	⊢ (((((𝜑 ∧ 𝜓) ∧ 𝜏) ∧ 𝜂) ∧ 𝜒) → 𝜃)

Proof of Theorem ad5ant125

Step	Hyp	Ref	Expression
1		ad5ant.1	. . . 4 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
2	1	3expia 1119	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃))
3	2	2a1d 26	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜏 → (𝜂 → (𝜒 → 𝜃))))
4	3	imp41 425	1 ⊢ (((((𝜑 ∧ 𝜓) ∧ 𝜏) ∧ 𝜂) ∧ 𝜒) → 𝜃)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1087

This theorem is referenced by:  supxrge  42767  hoidmvlelem3  44025