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Metamath Proof Explorer

Theorem el2122old

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

Ref Expression
Hypotheses el2122old.1 ⊢ (    (    𝜑    ,    𝜓    )    ▶    𝜒    )
el2122old.2 ⊢ (    𝜓    ▶    𝜃    )
el2122old.3 ⊢ (    𝜓    ▶    𝜏    )
el2122old.4 ⊢ ( ( 𝜒 ∧ 𝜃 ∧ 𝜏 ) → 𝜂 )
Assertion el2122old (    (    𝜑    ,    𝜓    )    ▶    𝜂    )

Proof

Step Hyp Ref Expression
1 el2122old.1 ⊢ (    (    𝜑    ,    𝜓    )    ▶    𝜒    )
2 el2122old.2 ⊢ (    𝜓    ▶    𝜃    )
3 el2122old.3 ⊢ (    𝜓    ▶    𝜏    )
4 el2122old.4 ⊢ ( ( 𝜒 ∧ 𝜃 ∧ 𝜏 ) → 𝜂 )
5 1 dfvd2ani ⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
6 2 in1 ⊢ ( 𝜓 → 𝜃 )
7 3 in1 ⊢ ( 𝜓 → 𝜏 )
8 5 6 7 4 eel2122old ⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜂 )
9 8 dfvd2anir ⊢ (    (    𝜑    ,    𝜓    )    ▶    𝜂    )