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Mirrors > Home > ILE Home > Th. List > bm1.3ii | Unicode version |
Description: Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 4046. Similar to Theorem 1.3ii of [BellMachover] p. 463. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
bm1.3ii.1 |
Ref | Expression |
---|---|
bm1.3ii |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bm1.3ii.1 | . . . . 5 | |
2 | elequ2 1691 | . . . . . . . 8 | |
3 | 2 | imbi2d 229 | . . . . . . 7 |
4 | 3 | albidv 1796 | . . . . . 6 |
5 | 4 | cbvexv 1890 | . . . . 5 |
6 | 1, 5 | mpbi 144 | . . . 4 |
7 | ax-sep 4046 | . . . 4 | |
8 | 6, 7 | pm3.2i 270 | . . 3 |
9 | 8 | exan 1671 | . 2 |
10 | 19.42v 1878 | . . . 4 | |
11 | bimsc1 947 | . . . . . 6 | |
12 | 11 | alanimi 1435 | . . . . 5 |
13 | 12 | eximi 1579 | . . . 4 |
14 | 10, 13 | sylbir 134 | . . 3 |
15 | 14 | exlimiv 1577 | . 2 |
16 | 9, 15 | ax-mp 5 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wal 1329 wex 1468 |
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-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-sep 4046 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: axpow3 4101 vpwex 4103 zfpair2 4132 axun2 4357 uniex2 4358 |
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