Theorem ad5ant135 1306

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

Hypothesis

Ref	Expression
ad5ant135.1	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Assertion

Ref	Expression
ad5ant135	⊢ (((((𝜑 ∧ 𝜏) ∧ 𝜓) ∧ 𝜂) ∧ 𝜒) → 𝜃)

Proof of Theorem ad5ant135

Step	Hyp	Ref	Expression
1		ad5ant135.1	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
2	1	3exp 1256	. . . . . . . . 9 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
3	2	a1ddd 78	. . . . . . . 8 ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜏 → 𝜃))))
4	3	a1ddd 78	. . . . . . 7 ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜂 → (𝜏 → 𝜃)))))
5	4	com45 95	. . . . . 6 ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜏 → (𝜂 → 𝜃)))))
6	5	com34 89	. . . . 5 ⊢ (𝜑 → (𝜓 → (𝜏 → (𝜒 → (𝜂 → 𝜃)))))
7	6	com23 84	. . . 4 ⊢ (𝜑 → (𝜏 → (𝜓 → (𝜒 → (𝜂 → 𝜃)))))
8	7	com45 95	. . 3 ⊢ (𝜑 → (𝜏 → (𝜓 → (𝜂 → (𝜒 → 𝜃)))))
9	8	imp 444	. 2 ⊢ ((𝜑 ∧ 𝜏) → (𝜓 → (𝜂 → (𝜒 → 𝜃))))
10	9	imp41 617	1 ⊢ (((((𝜑 ∧ 𝜏) ∧ 𝜓) ∧ 𝜂) ∧ 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:  supxrgelem  38494  hoicvr  39438  hspmbllem2  39517  smfaddlem1  39649