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Mirrors > Home > ILE Home > Th. List > ifeq1dadc | Unicode version |
Description: Conditional equality. (Contributed by Jim Kingdon, 1-Jan-2022.) |
Ref | Expression |
---|---|
ifeq1dadc.1 | |
ifeq1dadc.dc | DECID |
Ref | Expression |
---|---|
ifeq1dadc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifeq1dadc.1 | . . 3 | |
2 | 1 | ifeq1d 3459 | . 2 |
3 | iffalse 3452 | . . . 4 | |
4 | iffalse 3452 | . . . 4 | |
5 | 3, 4 | eqtr4d 2153 | . . 3 |
6 | 5 | adantl 275 | . 2 |
7 | ifeq1dadc.dc | . . 3 DECID | |
8 | exmiddc 806 | . . 3 DECID | |
9 | 7, 8 | syl 14 | . 2 |
10 | 2, 6, 9 | mpjaodan 772 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 682 DECID wdc 804 wceq 1316 cif 3444 |
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-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-rab 2402 df-v 2662 df-un 3045 df-if 3445 |
This theorem is referenced by: sumeq2 11096 isumss 11128 |
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