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Metamath Proof Explorer

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    )