| Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > nfafv | Structured version Visualization version GIF version | ||
| Description: Bound-variable hypothesis builder for function value, analogous to nffv 6916. To prove a deduction version of this analogous to nffvd 6918 is not easily possible because a deduction version of nfdfat 47139 cannot be shown easily. (Contributed by Alexander van der Vekens, 26-May-2017.) |
| Ref | Expression |
|---|---|
| nfafv.1 | ⊢ Ⅎ𝑥𝐹 |
| nfafv.2 | ⊢ Ⅎ𝑥𝐴 |
| Ref | Expression |
|---|---|
| nfafv | ⊢ Ⅎ𝑥(𝐹'''𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfafv2 47144 | . 2 ⊢ (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) | |
| 2 | nfafv.1 | . . . 4 ⊢ Ⅎ𝑥𝐹 | |
| 3 | nfafv.2 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
| 4 | 2, 3 | nfdfat 47139 | . . 3 ⊢ Ⅎ𝑥 𝐹 defAt 𝐴 |
| 5 | 2, 3 | nffv 6916 | . . 3 ⊢ Ⅎ𝑥(𝐹‘𝐴) |
| 6 | nfcv 2905 | . . 3 ⊢ Ⅎ𝑥V | |
| 7 | 4, 5, 6 | nfif 4556 | . 2 ⊢ Ⅎ𝑥if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) |
| 8 | 1, 7 | nfcxfr 2903 | 1 ⊢ Ⅎ𝑥(𝐹'''𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: Ⅎwnfc 2890 Vcvv 3480 ifcif 4525 ‘cfv 6561 defAt wdfat 47128 '''cafv 47129 |
| 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 |
| This theorem is referenced by: csbafv12g 47149 nfaov 47191 |
| Copyright terms: Public domain | W3C validator |