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Mirrors > Home > ILE Home > Th. List > ifbieq2d | Unicode version |
Description: Equivalence/equality deduction for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.) |
Ref | Expression |
---|---|
ifbieq2d.1 |
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ifbieq2d.2 |
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Ref | Expression |
---|---|
ifbieq2d |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifbieq2d.1 |
. . 3
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2 | 1 | ifbid 3570 |
. 2
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3 | ifbieq2d.2 |
. . 3
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4 | 3 | ifeq2d 3567 |
. 2
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5 | 2, 4 | eqtrd 2222 |
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-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-rab 2477 df-v 2754 df-un 3148 df-if 3550 |
This theorem is referenced by: difinfsnlem 7129 ctmlemr 7138 xnegeq 9859 xaddval 9877 iseqf1olemqval 10520 iseqf1olemqk 10527 seq3f1olemqsum 10533 exp3val 10556 gcdval 11995 gcdass 12051 lcmval 12098 lcmass 12120 pcval 12331 ennnfonelemj0 12455 ennnfonelemjn 12456 ennnfonelem0 12459 ennnfonelemp1 12460 ennnfonelemnn0 12476 mulgval 13079 znval 13949 lgsval 14883 lgsfvalg 14884 lgsval2lem 14889 nnsf 15233 peano4nninf 15234 peano3nninf 15235 exmidsbthr 15250 |
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