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Mirrors > Home > ILE Home > Th. List > ssoprab2 | Unicode version |
Description: Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2 4277. (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 1455 |
. . . . . . . 8
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4 | exim 1599 |
. . . . . . . 8
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5 | 3, 4 | syl 14 |
. . . . . . 7
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6 | 5 | alimi 1455 |
. . . . . 6
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7 | exim 1599 |
. . . . . 6
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8 | 6, 7 | syl 14 |
. . . . 5
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9 | 8 | alimi 1455 |
. . . 4
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10 | exim 1599 |
. . . 4
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11 | 9, 10 | syl 14 |
. . 3
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12 | 11 | ss2abdv 3230 |
. 2
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13 | df-oprab 5882 |
. 2
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14 | df-oprab 5882 |
. 2
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15 | 12, 13, 14 | 3sstr4g 3200 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 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-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-in 3137 df-ss 3144 df-oprab 5882 |
This theorem is referenced by: ssoprab2b 5935 |
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