Theorem ad5ant15 1303

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

Hypothesis

Ref	Expression
ad5ant15.1	⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Assertion

Ref	Expression
ad5ant15	⊢ (((((𝜑 ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜓) → 𝜒)

Proof of Theorem ad5ant15

Step	Hyp	Ref	Expression
1		ad5ant15.1	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
2	1	ex 450	. . . . . . . . 9 ⊢ (𝜑 → (𝜓 → 𝜒))
3	2	2a1dd 51	. . . . . . . 8 ⊢ (𝜑 → (𝜓 → (𝜃 → (𝜏 → 𝜒))))
4	3	a1ddd 80	. . . . . . 7 ⊢ (𝜑 → (𝜓 → (𝜃 → (𝜂 → (𝜏 → 𝜒)))))
5	4	com45 97	. . . . . 6 ⊢ (𝜑 → (𝜓 → (𝜃 → (𝜏 → (𝜂 → 𝜒)))))
6	5	com23 86	. . . . 5 ⊢ (𝜑 → (𝜃 → (𝜓 → (𝜏 → (𝜂 → 𝜒)))))
7	6	com34 91	. . . 4 ⊢ (𝜑 → (𝜃 → (𝜏 → (𝜓 → (𝜂 → 𝜒)))))
8	7	com45 97	. . 3 ⊢ (𝜑 → (𝜃 → (𝜏 → (𝜂 → (𝜓 → 𝜒)))))
9	8	imp 445	. 2 ⊢ ((𝜑 ∧ 𝜃) → (𝜏 → (𝜂 → (𝜓 → 𝜒))))
10	9	imp41 619	1 ⊢ (((((𝜑 ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜓) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 384

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

This theorem is referenced by:  breprexplemc  30710  mblfinlem2  33447  supxrgelem  39553  supxrge  39554  rexabslelem  39645  uzub  39658  smflimlem4  40982