Theorem nfafv

Description: Bound-variable hypothesis builder for function value, analogous to nffv . To prove a deduction version of this analogous to nffvd is not easily possible because a deduction version of nfdfat cannot be shown easily. (Contributed by Alexander van der Vekens, 26-May-2017)

	Ref	Expression
Hypotheses	nfafv.1	⊢ Ⅎ 𝑥 𝐹
	nfafv.2	⊢ Ⅎ 𝑥 𝐴
Assertion	nfafv	⊢ Ⅎ 𝑥 ( 𝐹 ''' 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		nfafv.1	⊢ Ⅎ 𝑥 𝐹
2		nfafv.2	⊢ Ⅎ 𝑥 𝐴
3		dfafv2	⊢ ( 𝐹 ''' 𝐴 ) = if ( 𝐹 defAt 𝐴 , ( 𝐹 ‘ 𝐴 ) , V )
4	1 2	nfdfat	⊢ Ⅎ 𝑥 𝐹 defAt 𝐴
5	1 2	nffv	⊢ Ⅎ 𝑥 ( 𝐹 ‘ 𝐴 )
6		nfcv	⊢ Ⅎ 𝑥 V
7	4 5 6	nfif	⊢ Ⅎ 𝑥 if ( 𝐹 defAt 𝐴 , ( 𝐹 ‘ 𝐴 ) , V )
8	3 7	nfcxfr	⊢ Ⅎ 𝑥 ( 𝐹 ''' 𝐴 )