|   | Mathbox for Alexander van der Vekens | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > Mathboxes > nfaov | Structured version Visualization version GIF version | ||
| Description: Bound-variable hypothesis builder for operation value, analogous to nfov 7461. To prove a deduction version of this analogous to nfovd 7460 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 47148). (Contributed by Alexander van der Vekens, 26-May-2017.) | 
| Ref | Expression | 
|---|---|
| nfaov.2 | ⊢ Ⅎ𝑥𝐴 | 
| nfaov.3 | ⊢ Ⅎ𝑥𝐹 | 
| nfaov.4 | ⊢ Ⅎ𝑥𝐵 | 
| Ref | Expression | 
|---|---|
| nfaov | ⊢ Ⅎ𝑥 ((𝐴𝐹𝐵)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-aov 47133 | . 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 47148 | . 2 ⊢ Ⅎ𝑥(𝐹'''〈𝐴, 𝐵〉) | 
| 7 | 1, 6 | nfcxfr 2903 | 1 ⊢ Ⅎ𝑥 ((𝐴𝐹𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: Ⅎwnfc 2890 〈cop 4632 '''cafv 47129 ((caov 47130 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-res 5697 df-iota 6514 df-fun 6563 df-fv 6569 df-aiota 47097 df-dfat 47131 df-afv 47132 df-aov 47133 | 
| This theorem is referenced by: csbaovg 47192 | 
| Copyright terms: Public domain | W3C validator |