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Metamath Proof Explorer

Theorem ee120

Description: Virtual deduction rule e120 without virtual deduction symbols. (Contributed by Alan Sare, 14-Jul-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ee120.1	⊢ ( 𝜑 → 𝜓 )
	ee120.2	⊢ ( 𝜑 → ( 𝜒 → 𝜃 ) )
	ee120.3	⊢ 𝜏
	ee120.4	⊢ ( 𝜓 → ( 𝜃 → ( 𝜏 → 𝜂 ) ) )
Assertion	ee120	⊢ ( 𝜑 → ( 𝜒 → 𝜂 ) )

Proof

Step	Hyp	Ref	Expression
1		ee120.1	⊢ ( 𝜑 → 𝜓 )
2		ee120.2	⊢ ( 𝜑 → ( 𝜒 → 𝜃 ) )
3		ee120.3	⊢ 𝜏
4		ee120.4	⊢ ( 𝜓 → ( 𝜃 → ( 𝜏 → 𝜂 ) ) )
5	1	a1d	⊢ ( 𝜑 → ( 𝜒 → 𝜓 ) )
6	3	a1i	⊢ ( 𝜒 → 𝜏 )
7	6	a1i	⊢ ( 𝜑 → ( 𝜒 → 𝜏 ) )
8	5 2 7 4	ee222	⊢ ( 𝜑 → ( 𝜒 → 𝜂 ) )