Metamath Proof Explorer


Theorem e222

Description: A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Jun-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses e222.1 (    𝜑    ,    𝜓    ▶    𝜒    )
e222.2 (    𝜑    ,    𝜓    ▶    𝜃    )
e222.3 (    𝜑    ,    𝜓    ▶    𝜏    )
e222.4 ( 𝜒 → ( 𝜃 → ( 𝜏𝜂 ) ) )
Assertion e222 (    𝜑    ,    𝜓    ▶    𝜂    )

Proof

Step Hyp Ref Expression
1 e222.1 (    𝜑    ,    𝜓    ▶    𝜒    )
2 e222.2 (    𝜑    ,    𝜓    ▶    𝜃    )
3 e222.3 (    𝜑    ,    𝜓    ▶    𝜏    )
4 e222.4 ( 𝜒 → ( 𝜃 → ( 𝜏𝜂 ) ) )
5 3 dfvd2i ( 𝜑 → ( 𝜓𝜏 ) )
6 5 imp ( ( 𝜑𝜓 ) → 𝜏 )
7 1 dfvd2i ( 𝜑 → ( 𝜓𝜒 ) )
8 7 imp ( ( 𝜑𝜓 ) → 𝜒 )
9 2 dfvd2i ( 𝜑 → ( 𝜓𝜃 ) )
10 9 imp ( ( 𝜑𝜓 ) → 𝜃 )
11 8 10 4 syl2im ( ( 𝜑𝜓 ) → ( ( 𝜑𝜓 ) → ( 𝜏𝜂 ) ) )
12 11 pm2.43i ( ( 𝜑𝜓 ) → ( 𝜏𝜂 ) )
13 6 12 syl5com ( ( 𝜑𝜓 ) → ( ( 𝜑𝜓 ) → 𝜂 ) )
14 13 pm2.43i ( ( 𝜑𝜓 ) → 𝜂 )
15 14 ex ( 𝜑 → ( 𝜓𝜂 ) )
16 15 dfvd2ir (    𝜑    ,    𝜓    ▶    𝜂    )