Theorem el123

Description: A virtual deduction elimination rule. (Contributed by Alan Sare, 13-Jun-2015) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	el123.1	⊢ (    𝜑    ▶    𝜓    )
	el123.2	⊢ (    𝜒    ▶    𝜃    )
	el123.3	⊢ (    𝜏    ▶    𝜂    )
	el123.4	⊢ ( ( 𝜓 ∧ 𝜃 ∧ 𝜂 ) → 𝜁 )
Assertion	el123	⊢ (    (    𝜑    ,    𝜒    ,    𝜏    )    ▶    𝜁    )

Proof

Step	Hyp	Ref	Expression
1		el123.1	⊢ (    𝜑    ▶    𝜓    )
2		el123.2	⊢ (    𝜒    ▶    𝜃    )
3		el123.3	⊢ (    𝜏    ▶    𝜂    )
4		el123.4	⊢ ( ( 𝜓 ∧ 𝜃 ∧ 𝜂 ) → 𝜁 )
5	1	in1	⊢ ( 𝜑 → 𝜓 )
6	2	in1	⊢ ( 𝜒 → 𝜃 )
7	3	in1	⊢ ( 𝜏 → 𝜂 )
8	5 6 7 4	syl3an	⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜏 ) → 𝜁 )
9	8	dfvd3anir	⊢ (    (    𝜑    ,    𝜒    ,    𝜏    )    ▶    𝜁    )