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 6198. To prove a deduction version of this analogous to nffvd 6200 is not easily possible because a deduction version of nfdfat 41210 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 41212 | . 2 ⊢ (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) | |
2 | nfafv.1 | . . . 4 ⊢ Ⅎ𝑥𝐹 | |
3 | nfafv.2 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
4 | 2, 3 | nfdfat 41210 | . . 3 ⊢ Ⅎ𝑥 𝐹 defAt 𝐴 |
5 | 2, 3 | nffv 6198 | . . 3 ⊢ Ⅎ𝑥(𝐹‘𝐴) |
6 | nfcv 2764 | . . 3 ⊢ Ⅎ𝑥V | |
7 | 4, 5, 6 | nfif 4115 | . 2 ⊢ Ⅎ𝑥if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) |
8 | 1, 7 | nfcxfr 2762 | 1 ⊢ Ⅎ𝑥(𝐹'''𝐴) |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnfc 2751 Vcvv 3200 ifcif 4086 ‘cfv 5888 defAt wdfat 41193 '''cafv 41194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-res 5126 df-iota 5851 df-fun 5890 df-fv 5896 df-dfat 41196 df-afv 41197 |
This theorem is referenced by: csbafv12g 41217 nfaov 41259 |
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