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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 6668. To prove a deduction version of this analogous to nffvd 6670 is not easily possible because a deduction version of nfdfat 44073 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 44078 | . 2 ⊢ (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) | |
2 | nfafv.1 | . . . 4 ⊢ Ⅎ𝑥𝐹 | |
3 | nfafv.2 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
4 | 2, 3 | nfdfat 44073 | . . 3 ⊢ Ⅎ𝑥 𝐹 defAt 𝐴 |
5 | 2, 3 | nffv 6668 | . . 3 ⊢ Ⅎ𝑥(𝐹‘𝐴) |
6 | nfcv 2919 | . . 3 ⊢ Ⅎ𝑥V | |
7 | 4, 5, 6 | nfif 4450 | . 2 ⊢ Ⅎ𝑥if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) |
8 | 1, 7 | nfcxfr 2917 | 1 ⊢ Ⅎ𝑥(𝐹'''𝐴) |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnfc 2899 Vcvv 3409 ifcif 4420 ‘cfv 6335 defAt wdfat 44062 '''cafv 44063 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-sep 5169 ax-nul 5176 ax-pow 5234 ax-pr 5298 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-ral 3075 df-rex 3076 df-rab 3079 df-v 3411 df-sbc 3697 df-csb 3806 df-dif 3861 df-un 3863 df-in 3865 df-ss 3875 df-nul 4226 df-if 4421 df-sn 4523 df-pr 4525 df-op 4529 df-uni 4799 df-int 4839 df-br 5033 df-opab 5095 df-id 5430 df-xp 5530 df-rel 5531 df-cnv 5532 df-co 5533 df-dm 5534 df-res 5536 df-iota 6294 df-fun 6337 df-fv 6343 df-aiota 44030 df-dfat 44065 df-afv 44066 |
This theorem is referenced by: csbafv12g 44083 nfaov 44125 |
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