Theorem nfaov 47944

Description: Bound-variable hypothesis builder for operation value, analogous to nfov 7442. To prove a deduction version of this analogous to nfovd 7441 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 47901). (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 47886	. 2 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
2		nfaov.3	. . 3 ⊢ Ⅎ𝑥𝐹
3		nfaov.2	. . . 4 ⊢ Ⅎ𝑥𝐴
4		nfaov.4	. . . 4 ⊢ Ⅎ𝑥𝐵
5	3, 4	nfop 4853	. . 3 ⊢ Ⅎ𝑥⟨𝐴, 𝐵⟩
6	2, 5	nfafv 47901	. 2 ⊢ Ⅎ𝑥(𝐹'''⟨𝐴, 𝐵⟩)
7	1, 6	nfcxfr 2922	1 ⊢ Ⅎ𝑥 ((𝐴𝐹𝐵))

This proof depends on syntax axioms:  Ⅎwnfc 2909  ⟨cop 4594  '''cafv 47882   ((caov 47883

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-br 5109  df-opab 5173  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-res 5672  df-iota 6492  df-fun 6538  df-fv 6544  df-aiota 47850  df-dfat 47884  df-afv 47885  df-aov 47886

This theorem is used by:  csbaovg  47945