Theorem ad5ant234 1300

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

Hypothesis

Ref	Expression
ad5ant234.1	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Assertion

Ref	Expression
ad5ant234	⊢ (((((𝜏 ∧ 𝜑) ∧ 𝜓) ∧ 𝜒) ∧ 𝜂) → 𝜃)

Proof of Theorem ad5ant234

Step	Hyp	Ref	Expression
1		ad5ant234.1	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
2	1	3exp 1256	. . . . . 6 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
3	2	a1ddd 78	. . . . 5 ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜏 → 𝜃))))
4	3	a1ddd 78	. . . 4 ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜂 → (𝜏 → 𝜃)))))
5	4	com5r 102	. . 3 ⊢ (𝜏 → (𝜑 → (𝜓 → (𝜒 → (𝜂 → 𝜃)))))
6	5	imp 444	. 2 ⊢ ((𝜏 ∧ 𝜑) → (𝜓 → (𝜒 → (𝜂 → 𝜃))))
7	6	imp41 617	1 ⊢ (((((𝜏 ∧ 𝜑) ∧ 𝜓) ∧ 𝜒) ∧ 𝜂) → 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: (None)