Theorem nfaov 48075

Description: Bound-variable hypothesis builder for operation value, analogous to nfov 7447. To prove a deduction version of this analogous to nfovd 7446 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 48032). (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 48017	. 2 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
2		nfaov.3	. . 3 ⊢ Ⅎ𝑥𝐹
3		nfaov.2	. . . 4 ⊢ Ⅎ𝑥𝐴
4		nfaov.4	. . . 4 ⊢ Ⅎ𝑥𝐵
5	3, 4	nfop 4852	. . 3 ⊢ Ⅎ𝑥⟨𝐴, 𝐵⟩
6	2, 5	nfafv 48032	. 2 ⊢ Ⅎ𝑥(𝐹'''⟨𝐴, 𝐵⟩)
7	1, 6	nfcxfr 2922	1 ⊢ Ⅎ𝑥 ((𝐴𝐹𝐵))

This proof depends on syntax axioms:  Ⅎwnfc 2909  ⟨cop 4593  '''cafv 48013   ((caov 48014

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-br 5108  df-opab 5172  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-res 5671  df-iota 6493  df-fun 6539  df-fv 6545  df-aiota 47981  df-dfat 48015  df-afv 48016  df-aov 48017

This theorem is used by:  csbaovg  48076