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Metamath Proof Explorer

Theorem vd23

Description: A virtual deduction with 2 virtual hypotheses virtually inferring a virtual conclusion infers that the same conclusion is virtually inferred by the same 2 virtual hypotheses and a third hypothesis. (Contributed by Alan Sare, 12-Jun-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypothesis vd23.1 ⊢ (    𝜑    ,    𝜓    ▶    𝜒    )
Assertion vd23 (    𝜑    ,    𝜓    ,    𝜃    ▶    𝜒    )

Proof

Step Hyp Ref Expression
1 vd23.1 ⊢ (    𝜑    ,    𝜓    ▶    𝜒    )
2 1 dfvd2i ⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
3 2 a1dd ⊢ ( 𝜑 → ( 𝜓 → ( 𝜃 → 𝜒 ) ) )
4 3 dfvd3ir ⊢ (    𝜑    ,    𝜓    ,    𝜃    ▶    𝜒    )