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Mirrors > Home > ILE Home > Th. List > eqifdc | Unicode version |
Description: Expansion of an equality with a conditional operator. (Contributed by Jim Kingdon, 28-Jul-2022.) |
Ref | Expression |
---|---|
eqifdc |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exmiddc 836 |
. . 3
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2 | simpr 110 |
. . . . . 6
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3 | simpl 109 |
. . . . . . 7
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4 | 2 | iftrued 3541 |
. . . . . . 7
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5 | 3, 4 | eqtrd 2210 |
. . . . . 6
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6 | 2, 5 | jca 306 |
. . . . 5
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7 | 6 | ex 115 |
. . . 4
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8 | simpr 110 |
. . . . . 6
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9 | simpl 109 |
. . . . . . 7
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10 | 8 | iffalsed 3544 |
. . . . . . 7
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11 | 9, 10 | eqtrd 2210 |
. . . . . 6
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12 | 8, 11 | jca 306 |
. . . . 5
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13 | 12 | ex 115 |
. . . 4
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14 | 7, 13 | orim12d 786 |
. . 3
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15 | 1, 14 | syl5com 29 |
. 2
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16 | simpr 110 |
. . . 4
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17 | simpl 109 |
. . . . 5
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18 | 17 | iftrued 3541 |
. . . 4
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19 | 16, 18 | eqtr4d 2213 |
. . 3
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20 | simpr 110 |
. . . 4
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21 | simpl 109 |
. . . . 5
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22 | 21 | iffalsed 3544 |
. . . 4
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23 | 20, 22 | eqtr4d 2213 |
. . 3
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24 | 19, 23 | jaoi 716 |
. 2
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25 | 15, 24 | impbid1 142 |
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: fodjum 7143 nninfwlporlemd 7169 xrmaxiflemcom 11252 subctctexmid 14632 |
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