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Metamath Proof Explorer

Theorem e1111

Description: A virtual deduction elimination rule. (Contributed by Alan Sare, 6-Mar-2012) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses e1111.1 ⊢ (    𝜑    ▶    𝜓    )
e1111.2 ⊢ (    𝜑    ▶    𝜒    )
e1111.3 ⊢ (    𝜑    ▶    𝜃    )
e1111.4 ⊢ (    𝜑    ▶    𝜏    )
e1111.5 ⊢ ( 𝜓 → ( 𝜒 → ( 𝜃 → ( 𝜏 → 𝜂 ) ) ) )
Assertion e1111 (    𝜑    ▶    𝜂    )

Proof

Step Hyp Ref Expression
1 e1111.1 ⊢ (    𝜑    ▶    𝜓    )
2 e1111.2 ⊢ (    𝜑    ▶    𝜒    )
3 e1111.3 ⊢ (    𝜑    ▶    𝜃    )
4 e1111.4 ⊢ (    𝜑    ▶    𝜏    )
5 e1111.5 ⊢ ( 𝜓 → ( 𝜒 → ( 𝜃 → ( 𝜏 → 𝜂 ) ) ) )
6 1 in1 ⊢ ( 𝜑 → 𝜓 )
7 2 in1 ⊢ ( 𝜑 → 𝜒 )
8 3 in1 ⊢ ( 𝜑 → 𝜃 )
9 4 in1 ⊢ ( 𝜑 → 𝜏 )
10 6 7 8 9 5 ee1111 ⊢ ( 𝜑 → 𝜂 )
11 10 dfvd1ir ⊢ (    𝜑    ▶    𝜂    )