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| Mirrors > Home > ILE Home > Th. List > ifbieq1d | Unicode version | ||
| Description: Equivalence/equality deduction for conditional operators. (Contributed by JJ, 25-Sep-2018.) |
| Ref | Expression |
|---|---|
| ifbieq1d.1 |
|
| ifbieq1d.2 |
|
| Ref | Expression |
|---|---|
| ifbieq1d |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ifbieq1d.1 |
. . 3
| |
| 2 | 1 | ifbid 3591 |
. 2
|
| 3 | ifbieq1d.2 |
. . 3
| |
| 4 | 3 | ifeq1d 3587 |
. 2
|
| 5 | 2, 4 | eqtrd 2237 |
1
|
| 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 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-ext 2186 |
| This theorem depends on definitions: df-bi 117 df-tru 1375 df-nf 1483 df-sb 1785 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-rab 2492 df-v 2773 df-un 3169 df-if 3571 |
| This theorem is referenced by: ctssdclemn0 7211 ctssdc 7214 enumctlemm 7215 iseqf1olemfvp 10653 seq3f1olemqsum 10656 seq3f1oleml 10659 seq3f1o 10660 bcval 10892 sumrbdclem 11659 summodclem3 11662 summodclem2a 11663 summodc 11665 zsumdc 11666 fsum3 11669 isumss 11673 isumss2 11675 fsum3cvg2 11676 fsum3ser 11679 fsumcl2lem 11680 fsumadd 11688 sumsnf 11691 fsummulc2 11730 isumlessdc 11778 cbvprod 11840 prodrbdclem 11853 prodmodclem3 11857 prodmodclem2a 11858 prodmodc 11860 zproddc 11861 fprodseq 11865 fprodntrivap 11866 prodssdc 11871 fprodmul 11873 prodsnf 11874 pcmpt 12637 pcmptdvds 12639 elply2 15178 lgsval 15452 lgsfvalg 15453 lgsdir 15483 lgsdilem2 15484 lgsdi 15485 lgsne0 15486 |
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