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Metamath Proof Explorer

Theorem nfif

Description: Bound-variable hypothesis builder for a conditional operator. (Contributed by NM, 16-Feb-2005) (Proof shortened by Andrew Salmon, 26-Jun-2011)

Ref Expression
Hypotheses nfif.1 ⊢ Ⅎ x φ
nfif.2 ⊢ Ⅎ _ x A
nfif.3 ⊢ Ⅎ _ x B
Assertion nfif ⊢ Ⅎ _ x if φ A B

Proof

Step Hyp Ref Expression
1 nfif.1 ⊢ Ⅎ x φ
2 nfif.2 ⊢ Ⅎ _ x A
3 nfif.3 ⊢ Ⅎ _ x B
4 1 a1i ⊢ ⊤ → Ⅎ x φ
5 2 a1i ⊢ ⊤ → Ⅎ _ x A
6 3 a1i ⊢ ⊤ → Ⅎ _ x B
7 4 5 6 nfifd ⊢ ⊤ → Ⅎ _ x if φ A B
8 7 mptru ⊢ Ⅎ _ x if φ A B