Theorem nfaov 44671

Description: Bound-variable hypothesis builder for operation value, analogous to nfov 7305. To prove a deduction version of this analogous to nfovd 7304 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 44628). (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 44613	. 2 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
2		nfaov.3	. . 3 ⊢ Ⅎ𝑥𝐹
3		nfaov.2	. . . 4 ⊢ Ⅎ𝑥𝐴
4		nfaov.4	. . . 4 ⊢ Ⅎ𝑥𝐵
5	3, 4	nfop 4820	. . 3 ⊢ Ⅎ𝑥⟨𝐴, 𝐵⟩
6	2, 5	nfafv 44628	. 2 ⊢ Ⅎ𝑥(𝐹'''⟨𝐴, 𝐵⟩)
7	1, 6	nfcxfr 2905	1 ⊢ Ⅎ𝑥 ((𝐴𝐹𝐵))

Syntax hints:  Ⅎwnfc 2887  ⟨cop 4567  '''cafv 44609   ((caov 44610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-res 5601  df-iota 6391  df-fun 6435  df-fv 6441  df-aiota 44577  df-dfat 44611  df-afv 44612  df-aov 44613

This theorem is referenced by:  csbaovg  44672