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Mirrors > Home > NFE Home > Th. List > ifbi | Unicode version |
Description: Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.) |
Ref | Expression |
---|---|
ifbi |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfbi3 863 |
. 2
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2 | iftrue 3669 |
. . . 4
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3 | iftrue 3669 |
. . . . 5
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4 | 3 | eqcomd 2358 |
. . . 4
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5 | 2, 4 | sylan9eq 2405 |
. . 3
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6 | iffalse 3670 |
. . . 4
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7 | iffalse 3670 |
. . . . 5
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8 | 7 | eqcomd 2358 |
. . . 4
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9 | 6, 8 | sylan9eq 2405 |
. . 3
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10 | 5, 9 | jaoi 368 |
. 2
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11 | 1, 10 | sylbi 187 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-if 3664 |
This theorem is referenced by: ifbid 3681 ifbieq2i 3682 |
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