Theorem ad5ant2345 1369

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

Hypothesis

Ref	Expression
ad5ant2345.1	⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)

Assertion

Ref	Expression
ad5ant2345	⊢ (((((𝜂 ∧ 𝜑) ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)

Proof of Theorem ad5ant2345

Step	Hyp	Ref	Expression
1		ad5ant2345.1	. . . 4 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)
2	1	exp41 435	. . 3 ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → 𝜏))))
3	2	adantl 482	. 2 ⊢ ((𝜂 ∧ 𝜑) → (𝜓 → (𝜒 → (𝜃 → 𝜏))))
4	3	imp41 426	1 ⊢ (((((𝜂 ∧ 𝜑) ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)

Syntax hints:   → wi 4   ∧ wa 396

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 206  df-an 397

This theorem is referenced by:  mblfinlem2  35815  liminflelimsuplem  43316  climxlim2lem  43386  iundjiun  43998