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Mirrors > Home > ILE Home > Th. List > sbim | Unicode version |
Description: Implication inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 3-Feb-2018.) |
Ref | Expression |
---|---|
sbim |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbimv 1814 | . . . 4 | |
2 | 1 | sbbii 1688 | . . 3 |
3 | sbimv 1814 | . . 3 | |
4 | 2, 3 | bitri 182 | . 2 |
5 | ax-17 1459 | . . 3 | |
6 | 5 | sbco2v 1862 | . 2 |
7 | ax-17 1459 | . . . 4 | |
8 | 7 | sbco2v 1862 | . . 3 |
9 | ax-17 1459 | . . . 4 | |
10 | 9 | sbco2v 1862 | . . 3 |
11 | 8, 10 | imbi12i 237 | . 2 |
12 | 4, 6, 11 | 3bitr3i 208 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 103 wsb 1685 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 |
This theorem depends on definitions: df-bi 115 df-nf 1390 df-sb 1686 |
This theorem is referenced by: sbrim 1871 sblim 1872 sbbi 1874 moimv 2007 nfraldya 2400 sbcimg 2855 zfregfr 4316 tfi 4323 peano2 4336 |
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