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Mirrors > Home > ILE Home > Th. List > ifcldcd | Unicode version |
Description: Membership (closure) of a conditional operator, deduction form. (Contributed by Jim Kingdon, 8-Aug-2021.) |
Ref | Expression |
---|---|
ifcldcd.a |
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ifcldcd.b |
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ifcldcd.dc |
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Ref | Expression |
---|---|
ifcldcd |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iftrue 3539 |
. . . 4
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2 | 1 | adantl 277 |
. . 3
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3 | ifcldcd.a |
. . . 4
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4 | 3 | adantr 276 |
. . 3
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5 | 2, 4 | eqeltrd 2254 |
. 2
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6 | iffalse 3542 |
. . . 4
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7 | 6 | adantl 277 |
. . 3
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8 | ifcldcd.b |
. . . 4
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9 | 8 | adantr 276 |
. . 3
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10 | 7, 9 | eqeltrd 2254 |
. 2
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11 | ifcldcd.dc |
. . 3
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12 | df-dc 835 |
. . 3
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13 | 11, 12 | sylib 122 |
. 2
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14 | 5, 10, 13 | mpjaodan 798 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-11 1506 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-if 3535 |
This theorem is referenced by: fimax2gtrilemstep 6899 nnnninf 7123 nnnninfeq 7125 fodjuf 7142 fodjum 7143 fodju0 7144 mkvprop 7155 nninfwlporlemd 7169 nninfwlporlem 7170 nninfwlpoimlemg 7172 nninfwlpoimlemginf 7173 xaddf 9842 xaddval 9843 uzin2 10991 fsum3ser 11400 fsumsplit 11410 explecnv 11508 fprodsplitdc 11599 pcmpt2 12336 ennnfonelemp1 12401 opifismgmdc 12784 lgsval 14336 lgsfvalg 14337 lgsfcl2 14338 lgscllem 14339 lgsval2lem 14342 lgsneg 14356 lgsdilem 14359 lgsdir2 14365 lgsdir 14367 lgsdi 14369 lgsne0 14370 bj-charfundc 14480 nnsf 14674 peano4nninf 14675 nninfsellemcl 14680 nninffeq 14689 dceqnconst 14727 dcapnconst 14728 |
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