Mathbox for Alexander van der Vekens |
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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 6110. To prove a deduction version of this analogous to nffvd 6112 is not easily possible because a deduction version of nfdfat 39859 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 39861 | . 2 ⊢ (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) | |
2 | nfafv.1 | . . . 4 ⊢ Ⅎ𝑥𝐹 | |
3 | nfafv.2 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
4 | 2, 3 | nfdfat 39859 | . . 3 ⊢ Ⅎ𝑥 𝐹 defAt 𝐴 |
5 | 2, 3 | nffv 6110 | . . 3 ⊢ Ⅎ𝑥(𝐹‘𝐴) |
6 | nfcv 2751 | . . 3 ⊢ Ⅎ𝑥V | |
7 | 4, 5, 6 | nfif 4065 | . 2 ⊢ Ⅎ𝑥if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) |
8 | 1, 7 | nfcxfr 2749 | 1 ⊢ Ⅎ𝑥(𝐹'''𝐴) |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnfc 2738 Vcvv 3173 ifcif 4036 ‘cfv 5804 defAt wdfat 39842 '''cafv 39843 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-res 5050 df-iota 5768 df-fun 5806 df-fv 5812 df-dfat 39845 df-afv 39846 |
This theorem is referenced by: csbafv12g 39866 nfaov 39908 |
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