Theorem e223

Description: A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Dec-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	e223.1	⊢ (    𝜑    ,    𝜓    ▶    𝜒    )
	e223.2	⊢ (    𝜑    ,    𝜓    ▶    𝜃    )
	e223.3	⊢ (    𝜑    ,    𝜓    ,    𝜏    ▶    𝜂    )
	e223.4	⊢ ( 𝜒 → ( 𝜃 → ( 𝜂 → 𝜁 ) ) )
Assertion	e223	⊢ (    𝜑    ,    𝜓    ,    𝜏    ▶    𝜁    )

Proof

Step	Hyp	Ref	Expression
1		e223.1	⊢ (    𝜑    ,    𝜓    ▶    𝜒    )
2		e223.2	⊢ (    𝜑    ,    𝜓    ▶    𝜃    )
3		e223.3	⊢ (    𝜑    ,    𝜓    ,    𝜏    ▶    𝜂    )
4		e223.4	⊢ ( 𝜒 → ( 𝜃 → ( 𝜂 → 𝜁 ) ) )
5	1	in2	⊢ (    𝜑    ▶    ( 𝜓 → 𝜒 )    )
6	5	in1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
7	2	in2	⊢ (    𝜑    ▶    ( 𝜓 → 𝜃 )    )
8	7	in1	⊢ ( 𝜑 → ( 𝜓 → 𝜃 ) )
9	3	in3	⊢ (    𝜑    ,    𝜓    ▶    ( 𝜏 → 𝜂 )    )
10	9	in2	⊢ (    𝜑    ▶    ( 𝜓 → ( 𝜏 → 𝜂 ) )    )
11	10	in1	⊢ ( 𝜑 → ( 𝜓 → ( 𝜏 → 𝜂 ) ) )
12	6 8 11 4	ee223	⊢ ( 𝜑 → ( 𝜓 → ( 𝜏 → 𝜁 ) ) )
13	12	dfvd3ir	⊢ (    𝜑    ,    𝜓    ,    𝜏    ▶    𝜁    )