Theorem e12an 45474

Description: Conjunction form of e12 45473 (see syl6an 697). (Contributed by Alan Sare, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
e12an.1	⊢ (   𝜑   ▶   𝜓   )
e12an.2	⊢ (   𝜑   ,   𝜒   ▶   𝜃   )
e12an.3	⊢ ((𝜓 ∧ 𝜃) → 𝜏)

Assertion

Ref	Expression
e12an	⊢ (   𝜑   ,   𝜒   ▶   𝜏   )

Proof of Theorem e12an

Step	Hyp	Ref	Expression
1		e12an.1	. 2 ⊢ (   𝜑   ▶   𝜓   )
2		e12an.2	. 2 ⊢ (   𝜑   ,   𝜒   ▶   𝜃   )
3		e12an.3	. . 3 ⊢ ((𝜓 ∧ 𝜃) → 𝜏)
4	3	ex 418	. 2 ⊢ (𝜓 → (𝜃 → 𝜏))
5	1, 2, 4	e12 45473	1 ⊢ (   𝜑   ,   𝜒   ▶   𝜏   )

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  (   wvd1 45319  (   wvd2 45327

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-vd1 45320  df-vd2 45328

This theorem is used by:  sstrALT2VD  45583