Theorem ad5ant145 1307

Description: Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)

Hypothesis

Ref	Expression
ad5ant145.1	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Assertion

Ref	Expression
ad5ant145	⊢ (((((𝜑 ∧ 𝜏) ∧ 𝜂) ∧ 𝜓) ∧ 𝜒) → 𝜃)

Proof of Theorem ad5ant145

Step	Hyp	Ref	Expression
1		ad5ant145.1	. . . . 5 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
2	1	3exp 1256	. . . 4 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
3	2	2a1d 26	. . 3 ⊢ (𝜑 → (𝜏 → (𝜂 → (𝜓 → (𝜒 → 𝜃)))))
4	3	imp 444	. 2 ⊢ ((𝜑 ∧ 𝜏) → (𝜂 → (𝜓 → (𝜒 → 𝜃))))
5	4	imp41 617	1 ⊢ (((((𝜑 ∧ 𝜏) ∧ 𝜂) ∧ 𝜓) ∧ 𝜒) → 𝜃)

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031

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 196  df-an 385  df-3an 1033

This theorem is referenced by:  matunitlindflem1  32575  hspmbllem2  39517  smflimlem2  39658