Theorem ad5ant245 1307

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

Hypothesis

Ref	Expression
ad5ant245.1	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Assertion

Ref	Expression
ad5ant245	⊢ (((((𝜏 ∧ 𝜑) ∧ 𝜂) ∧ 𝜓) ∧ 𝜒) → 𝜃)

Proof of Theorem ad5ant245

Step	Hyp	Ref	Expression
1		ad5ant245.1	. . . . 5 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
2	1	3exp 1264	. . . 4 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
3	2	a1i13 27	. . 3 ⊢ (𝜏 → (𝜑 → (𝜂 → (𝜓 → (𝜒 → 𝜃)))))
4	3	imp 445	. 2 ⊢ ((𝜏 ∧ 𝜑) → (𝜂 → (𝜓 → (𝜒 → 𝜃))))
5	4	imp41 619	1 ⊢ (((((𝜏 ∧ 𝜑) ∧ 𝜂) ∧ 𝜓) ∧ 𝜒) → 𝜃)

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037

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 197  df-an 386  df-3an 1039

This theorem is referenced by:  2pthnloop  26627  matunitlindflem1  33405  nnfoctbdjlem  40672  sfprmdvdsmersenne  41520