Theorem nfaov 46186

Description: Bound-variable hypothesis builder for operation value, analogous to nfov 7441. To prove a deduction version of this analogous to nfovd 7440 is not quickly possible because many deduction versions for bound-variable hypothesis builder for constructs the definition of alternative operation values is based on are not available (see nfafv 46143). (Contributed by Alexander van der Vekens, 26-May-2017.)

Hypotheses

Ref	Expression
nfaov.2	⊢ Ⅎ𝑥𝐴
nfaov.3	⊢ Ⅎ𝑥𝐹
nfaov.4	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
nfaov	⊢ Ⅎ𝑥 ((𝐴𝐹𝐵))

Proof of Theorem nfaov

Step	Hyp	Ref	Expression
1		df-aov 46128	. 2 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
2		nfaov.3	. . 3 ⊢ Ⅎ𝑥𝐹
3		nfaov.2	. . . 4 ⊢ Ⅎ𝑥𝐴
4		nfaov.4	. . . 4 ⊢ Ⅎ𝑥𝐵
5	3, 4	nfop 4889	. . 3 ⊢ Ⅎ𝑥⟨𝐴, 𝐵⟩
6	2, 5	nfafv 46143	. 2 ⊢ Ⅎ𝑥(𝐹'''⟨𝐴, 𝐵⟩)
7	1, 6	nfcxfr 2901	1 ⊢ Ⅎ𝑥 ((𝐴𝐹𝐵))

Syntax hints:  Ⅎwnfc 2883  ⟨cop 4634  '''cafv 46124   ((caov 46125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-res 5688  df-iota 6495  df-fun 6545  df-fv 6551  df-aiota 46092  df-dfat 46126  df-afv 46127  df-aov 46128

This theorem is referenced by:  csbaovg  46187