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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 1880 | . . . 4 | |
2 | 1 | sbbii 1752 | . . 3 |
3 | sbimv 1880 | . . 3 | |
4 | 2, 3 | bitri 183 | . 2 |
5 | ax-17 1513 | . . 3 | |
6 | 5 | sbco2vh 1932 | . 2 |
7 | ax-17 1513 | . . . 4 | |
8 | 7 | sbco2vh 1932 | . . 3 |
9 | ax-17 1513 | . . . 4 | |
10 | 9 | sbco2vh 1932 | . . 3 |
11 | 8, 10 | imbi12i 238 | . 2 |
12 | 4, 6, 11 | 3bitr3i 209 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wsb 1749 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 |
This theorem depends on definitions: df-bi 116 df-nf 1448 df-sb 1750 |
This theorem is referenced by: sbrim 1943 sblim 1944 sbbi 1946 moimv 2079 nfraldya 2499 sbcimg 2987 zfregfr 4545 tfi 4553 peano2 4566 |
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