Theorem e10an 42315

Description: Conjunction form of e10 42314. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
e10an.1	⊢ (   𝜑   ▶   𝜓   )
e10an.2	⊢ 𝜒
e10an.3	⊢ ((𝜓 ∧ 𝜒) → 𝜃)

Assertion

Ref	Expression
e10an	⊢ (   𝜑   ▶   𝜃   )

Proof of Theorem e10an

Step	Hyp	Ref	Expression
1		e10an.1	. 2 ⊢ (   𝜑   ▶   𝜓   )
2		e10an.2	. 2 ⊢ 𝜒
3		e10an.3	. . 3 ⊢ ((𝜓 ∧ 𝜒) → 𝜃)
4	3	ex 413	. 2 ⊢ (𝜓 → (𝜒 → 𝜃))
5	1, 2, 4	e10 42314	1 ⊢ (   𝜑   ▶   𝜃   )

Syntax hints:   → wi 4   ∧ wa 396  (   wvd1 42189

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  df-vd1 42190

This theorem is referenced by:  snsslVD  42449