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Metamath Proof Explorer

Theorem nfov

Description: Bound-variable hypothesis builder for operation value. (Contributed by NM, 4-May-2004)

Ref Expression
Hypotheses nfov.1 ⊢ Ⅎ 𝑥 𝐴
nfov.2 ⊢ Ⅎ 𝑥 𝐹
nfov.3 ⊢ Ⅎ 𝑥 𝐵
Assertion nfov Ⅎ 𝑥 ( 𝐴 𝐹 𝐵 )

Proof

Step Hyp Ref Expression
1 nfov.1 ⊢ Ⅎ 𝑥 𝐴
2 nfov.2 ⊢ Ⅎ 𝑥 𝐹
3 nfov.3 ⊢ Ⅎ 𝑥 𝐵
4 1 a1i ⊢ ( ⊤ → Ⅎ 𝑥 𝐴 )
5 2 a1i ⊢ ( ⊤ → Ⅎ 𝑥 𝐹 )
6 3 a1i ⊢ ( ⊤ → Ⅎ 𝑥 𝐵 )
7 4 5 6 nfovd ⊢ ( ⊤ → Ⅎ 𝑥 ( 𝐴 𝐹 𝐵 ) )
8 7 mptru ⊢ Ⅎ 𝑥 ( 𝐴 𝐹 𝐵 )