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Metamath Proof Explorer

Theorem ee222

Description: e222 without virtual deduction connectives. Special theorem needed for the Virtual Deduction translation tool. (Contributed by Alan Sare, 7-Jul-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ee222.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
	ee222.2	⊢ ( 𝜑 → ( 𝜓 → 𝜃 ) )
	ee222.3	⊢ ( 𝜑 → ( 𝜓 → 𝜏 ) )
	ee222.4	⊢ ( 𝜒 → ( 𝜃 → ( 𝜏 → 𝜂 ) ) )
Assertion	ee222	⊢ ( 𝜑 → ( 𝜓 → 𝜂 ) )

Proof

Step	Hyp	Ref	Expression
1		ee222.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2		ee222.2	⊢ ( 𝜑 → ( 𝜓 → 𝜃 ) )
3		ee222.3	⊢ ( 𝜑 → ( 𝜓 → 𝜏 ) )
4		ee222.4	⊢ ( 𝜒 → ( 𝜃 → ( 𝜏 → 𝜂 ) ) )
5	1	imp	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
6	2	imp	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜃 )
7	3	imp	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜏 )
8	5 6 7 4	syl3c	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜂 )
9	8	ex	⊢ ( 𝜑 → ( 𝜓 → 𝜂 ) )