Theorem ee31an

Description: e31an without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ee31an.1	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) )
	ee31an.2	⊢ ( 𝜑 → 𝜏 )
	ee31an.3	⊢ ( ( 𝜃 ∧ 𝜏 ) → 𝜂 )
Assertion	ee31an	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜂 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ee31an.1	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) )
2		ee31an.2	⊢ ( 𝜑 → 𝜏 )
3		ee31an.3	⊢ ( ( 𝜃 ∧ 𝜏 ) → 𝜂 )
4	2	a1d	⊢ ( 𝜑 → ( 𝜒 → 𝜏 ) )
5	4	a1d	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜏 ) ) )
6	1 5 3	ee33an	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜂 ) ) )