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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 4064 | . 2 ⊢ (⊤ → Ⅎ𝑥if(𝜑, 𝐴, 𝐵)) |
8 | 7 | trud 1484 | 1 ⊢ Ⅎ𝑥if(𝜑, 𝐴, 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ⊤wtru 1476 Ⅎwnf 1699 Ⅎwnfc 2738 ifcif 4036 |
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-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-if 4037 |
This theorem is referenced by: csbif 4088 nfop 4356 nfrdg 7397 boxcutc 7837 nfoi 8302 nfsum1 14268 nfsum 14269 summolem2a 14293 zsum 14296 sumss 14302 sumss2 14304 fsumcvg2 14305 nfcprod 14480 cbvprod 14484 prodmolem2a 14503 zprod 14506 fprod 14510 fprodntriv 14511 prodss 14516 pcmpt 15434 pcmptdvds 15436 gsummpt1n0 18187 madugsum 20268 mbfpos 23224 mbfposb 23226 i1fposd 23280 isibl2 23339 nfitg 23347 cbvitg 23348 itgss3 23387 itgcn 23415 limcmpt 23453 rlimcnp2 24493 chirred 28638 cdleme31sn 34686 cdleme32d 34750 cdleme32f 34752 refsum2cn 38220 ssfiunibd 38464 icccncfext 38773 fourierdlem86 39085 vonicc 39576 nfafv 39865 |
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