Theorem e222

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

	Ref	Expression
Hypotheses	e222.1	⊢ (    𝜑    ,    𝜓    ▶    𝜒    )
	e222.2	⊢ (    𝜑    ,    𝜓    ▶    𝜃    )
	e222.3	⊢ (    𝜑    ,    𝜓    ▶    𝜏    )
	e222.4	⊢ ( 𝜒 → ( 𝜃 → ( 𝜏 → 𝜂 ) ) )
Assertion	e222	⊢ (    𝜑    ,    𝜓    ▶    𝜂    )

Proof

Step	Hyp	Ref	Expression
1		e222.1	⊢ (    𝜑    ,    𝜓    ▶    𝜒    )
2		e222.2	⊢ (    𝜑    ,    𝜓    ▶    𝜃    )
3		e222.3	⊢ (    𝜑    ,    𝜓    ▶    𝜏    )
4		e222.4	⊢ ( 𝜒 → ( 𝜃 → ( 𝜏 → 𝜂 ) ) )
5	3	dfvd2i	⊢ ( 𝜑 → ( 𝜓 → 𝜏 ) )
6	5	imp	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜏 )
7	1	dfvd2i	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
8	7	imp	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
9	2	dfvd2i	⊢ ( 𝜑 → ( 𝜓 → 𝜃 ) )
10	9	imp	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜃 )
11	8 10 4	syl2im	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( ( 𝜑 ∧ 𝜓 ) → ( 𝜏 → 𝜂 ) ) )
12	11	pm2.43i	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜏 → 𝜂 ) )
13	6 12	syl5com	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( ( 𝜑 ∧ 𝜓 ) → 𝜂 ) )
14	13	pm2.43i	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜂 )
15	14	ex	⊢ ( 𝜑 → ( 𝜓 → 𝜂 ) )
16	15	dfvd2ir	⊢ (    𝜑    ,    𝜓    ▶    𝜂    )