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Mirrors > Home > ILE Home > Th. List > sbcnestgf | Unicode version |
Description: Nest the composition of two substitutions. (Contributed by Mario Carneiro, 11-Nov-2016.) |
Ref | Expression |
---|---|
sbcnestgf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsbcq 2948 | . . . . 5 | |
2 | csbeq1 3043 | . . . . . 6 | |
3 | dfsbcq 2948 | . . . . . 6 | |
4 | 2, 3 | syl 14 | . . . . 5 |
5 | 1, 4 | bibi12d 234 | . . . 4 |
6 | 5 | imbi2d 229 | . . 3 |
7 | vex 2724 | . . . . 5 | |
8 | 7 | a1i 9 | . . . 4 |
9 | csbeq1a 3049 | . . . . . 6 | |
10 | dfsbcq 2948 | . . . . . 6 | |
11 | 9, 10 | syl 14 | . . . . 5 |
12 | 11 | adantl 275 | . . . 4 |
13 | nfnf1 1531 | . . . . 5 | |
14 | 13 | nfal 1563 | . . . 4 |
15 | nfa1 1528 | . . . . 5 | |
16 | nfcsb1v 3073 | . . . . . 6 | |
17 | 16 | a1i 9 | . . . . 5 |
18 | sp 1498 | . . . . 5 | |
19 | 15, 17, 18 | nfsbcd 2965 | . . . 4 |
20 | 8, 12, 14, 19 | sbciedf 2981 | . . 3 |
21 | 6, 20 | vtoclg 2781 | . 2 |
22 | 21 | imp 123 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wal 1340 wceq 1342 wnf 1447 wcel 2135 wnfc 2293 cvv 2721 wsbc 2946 csb 3040 |
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-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-sbc 2947 df-csb 3041 |
This theorem is referenced by: csbnestgf 3092 sbcnestg 3093 |
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