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Mirrors > Home > ILE Home > Th. List > ssoprab2 | Unicode version |
Description: Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2 4287. (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 337 |
. . . . . . . . 9
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3 | 2 | alimi 1465 |
. . . . . . . 8
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4 | exim 1609 |
. . . . . . . 8
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5 | 3, 4 | syl 14 |
. . . . . . 7
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6 | 5 | alimi 1465 |
. . . . . 6
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7 | exim 1609 |
. . . . . 6
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8 | 6, 7 | syl 14 |
. . . . 5
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9 | 8 | alimi 1465 |
. . . 4
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10 | exim 1609 |
. . . 4
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11 | 9, 10 | syl 14 |
. . 3
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12 | 11 | ss2abdv 3240 |
. 2
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13 | df-oprab 5892 |
. 2
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14 | df-oprab 5892 |
. 2
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15 | 12, 13, 14 | 3sstr4g 3210 |
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-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-in 3147 df-ss 3154 df-oprab 5892 |
This theorem is referenced by: ssoprab2b 5945 |
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