Metamath Proof Explorer


Theorem e13an

Description: A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses e13an.1 (    𝜑    ▶    𝜓    )
e13an.2 (    𝜑    ,    𝜒    ,    𝜃    ▶    𝜏    )
e13an.3 ( ( 𝜓𝜏 ) → 𝜂 )
Assertion e13an (    𝜑    ,    𝜒    ,    𝜃    ▶    𝜂    )

Proof

Step Hyp Ref Expression
1 e13an.1 (    𝜑    ▶    𝜓    )
2 e13an.2 (    𝜑    ,    𝜒    ,    𝜃    ▶    𝜏    )
3 e13an.3 ( ( 𝜓𝜏 ) → 𝜂 )
4 3 ex ( 𝜓 → ( 𝜏𝜂 ) )
5 1 2 4 e13 (    𝜑    ,    𝜒    ,    𝜃    ▶    𝜂    )