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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 )