Theorem nfaov 43372

Description: Bound-variable hypothesis builder for operation value, analogous to nfov 7180. To prove a deduction version of this analogous to nfovd 7179 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 43329). (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 43314	. 2 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
2		nfaov.3	. . 3 ⊢ Ⅎ𝑥𝐹
3		nfaov.2	. . . 4 ⊢ Ⅎ𝑥𝐴
4		nfaov.4	. . . 4 ⊢ Ⅎ𝑥𝐵
5	3, 4	nfop 4812	. . 3 ⊢ Ⅎ𝑥⟨𝐴, 𝐵⟩
6	2, 5	nfafv 43329	. 2 ⊢ Ⅎ𝑥(𝐹'''⟨𝐴, 𝐵⟩)
7	1, 6	nfcxfr 2975	1 ⊢ Ⅎ𝑥 ((𝐴𝐹𝐵))

Syntax hints:  Ⅎwnfc 2961  ⟨cop 4566  '''cafv 43310   ((caov 43311

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-int 4869  df-br 5059  df-opab 5121  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-res 5561  df-iota 6308  df-fun 6351  df-fv 6357  df-aiota 43279  df-dfat 43312  df-afv 43313  df-aov 43314

This theorem is referenced by:  csbaovg  43373