Theorem ee10an 44716

Description: e10an 44715 without virtual deductions. sylancl 586 is also e10an 44715 without virtual deductions, except the order of the hypotheses is different. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ee10an.1	⊢ (𝜑 → 𝜓)
ee10an.2	⊢ 𝜒
ee10an.3	⊢ ((𝜓 ∧ 𝜒) → 𝜃)

Assertion

Ref	Expression
ee10an	⊢ (𝜑 → 𝜃)

Proof of Theorem ee10an

Step	Hyp	Ref	Expression
1		ee10an.1	. 2 ⊢ (𝜑 → 𝜓)
2		ee10an.2	. 2 ⊢ 𝜒
3		ee10an.3	. 2 ⊢ ((𝜓 ∧ 𝜒) → 𝜃)
4	1, 2, 3	sylancl 586	1 ⊢ (𝜑 → 𝜃)

Syntax hints:   → wi 4   ∧ wa 395

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 207  df-an 396

This theorem is referenced by: (None)