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Mirrors > Home > ILE Home > Th. List > nfif | 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 9 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝜑) |
3 | nfif.2 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
4 | 3 | a1i 9 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝐴) |
5 | nfif.3 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
6 | 5 | a1i 9 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝐵) |
7 | 2, 4, 6 | nfifd 3494 | . 2 ⊢ (⊤ → Ⅎ𝑥if(𝜑, 𝐴, 𝐵)) |
8 | 7 | mptru 1340 | 1 ⊢ Ⅎ𝑥if(𝜑, 𝐴, 𝐵) |
Colors of variables: wff set class |
Syntax hints: ⊤wtru 1332 Ⅎwnf 1436 Ⅎwnfc 2266 ifcif 3469 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-if 3470 |
This theorem is referenced by: nfsum1 11118 nfsum 11119 sumrbdclem 11138 summodclem2a 11143 zsumdc 11146 fsum3 11149 isumss 11153 isumss2 11155 fsum3cvg2 11156 nfcprod1 11316 nfcprod 11317 cbvprod 11320 prodrbdclem 11333 |
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