Theorem e21an 42240

Description: Conjunction form of e21 42239. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
e21an.1	⊢ (   𝜑   ,   𝜓   ▶   𝜒   )
e21an.2	⊢ (   𝜑   ▶   𝜃   )
e21an.3	⊢ ((𝜒 ∧ 𝜃) → 𝜏)

Assertion

Ref	Expression
e21an	⊢ (   𝜑   ,   𝜓   ▶   𝜏   )

Proof of Theorem e21an

Step	Hyp	Ref	Expression
1		e21an.1	. 2 ⊢ (   𝜑   ,   𝜓   ▶   𝜒   )
2		e21an.2	. 2 ⊢ (   𝜑   ▶   𝜃   )
3		e21an.3	. . 3 ⊢ ((𝜒 ∧ 𝜃) → 𝜏)
4	3	ex 412	. 2 ⊢ (𝜒 → (𝜃 → 𝜏))
5	1, 2, 4	e21 42239	1 ⊢ (   𝜑   ,   𝜓   ▶   𝜏   )

Syntax hints:   → wi 4   ∧ wa 395  (   wvd1 42078  (   wvd2 42086

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 396  df-vd1 42079  df-vd2 42087

This theorem is referenced by: (None)