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Mirrors > Home > ILE Home > Th. List > ssoprab2 | Unicode version |
Description: Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2 4205. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 11-Dec-2016.) |
Ref | Expression |
---|---|
ssoprab2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 |
. . . . . . . . . 10
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2 | 1 | anim2d 335 |
. . . . . . . . 9
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3 | 2 | alimi 1432 |
. . . . . . . 8
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4 | exim 1579 |
. . . . . . . 8
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5 | 3, 4 | syl 14 |
. . . . . . 7
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6 | 5 | alimi 1432 |
. . . . . 6
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7 | exim 1579 |
. . . . . 6
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8 | 6, 7 | syl 14 |
. . . . 5
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9 | 8 | alimi 1432 |
. . . 4
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10 | exim 1579 |
. . . 4
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11 | 9, 10 | syl 14 |
. . 3
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12 | 11 | ss2abdv 3175 |
. 2
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13 | df-oprab 5786 |
. 2
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14 | df-oprab 5786 |
. 2
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15 | 12, 13, 14 | 3sstr4g 3145 |
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 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-in 3082 df-ss 3089 df-oprab 5786 |
This theorem is referenced by: ssoprab2b 5836 |
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