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Mirrors > Home > MPE Home > Th. List > nfif | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for a conditional operator. (Contributed by NM, 16-Feb-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
nfif.1 | ⊢ Ⅎ𝑥𝜑 |
nfif.2 | ⊢ Ⅎ𝑥𝐴 |
nfif.3 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
nfif | ⊢ Ⅎ𝑥if(𝜑, 𝐴, 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfif.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | a1i 11 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝜑) |
3 | nfif.2 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
4 | 3 | a1i 11 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝐴) |
5 | nfif.3 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
6 | 5 | a1i 11 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝐵) |
7 | 2, 4, 6 | nfifd 4558 | . 2 ⊢ (⊤ → Ⅎ𝑥if(𝜑, 𝐴, 𝐵)) |
8 | 7 | mptru 1549 | 1 ⊢ Ⅎ𝑥if(𝜑, 𝐴, 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ⊤wtru 1543 Ⅎwnf 1786 Ⅎwnfc 2884 ifcif 4529 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-if 4530 |
This theorem is referenced by: csbif 4586 nfop 4890 nfrdg 8414 boxcutc 8935 nfoi 9509 nfsum1 15636 nfsum 15637 summolem2a 15661 zsum 15664 sumss 15670 sumss2 15672 fsumcvg2 15673 nfcprod 15855 cbvprod 15859 prodmolem2a 15878 zprod 15881 fprod 15885 fprodntriv 15886 prodss 15891 pcmpt 16825 pcmptdvds 16827 gsummpt1n0 19833 madugsum 22145 mbfpos 25168 mbfposb 25170 i1fposd 25225 isibl2 25284 nfitg 25292 cbvitg 25293 itgss3 25332 itgcn 25362 limcmpt 25400 rlimcnp2 26471 nosupbnd2 27219 noinfbnd2 27234 chirred 31648 cdleme31sn 39251 cdleme32d 39315 cdleme32f 39317 refsum2cn 43722 ssfiunibd 44019 uzub 44141 limsupubuz 44429 icccncfext 44603 fourierdlem86 44908 vonicc 45401 nfafv 45844 nfafv2 45926 |
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