Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > sbcfung | Unicode version |
Description: Distribute proper substitution through the function predicate. (Contributed by Alexander van der Vekens, 23-Jul-2017.) |
Ref | Expression |
---|---|
sbcfung |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcan 2856 | . . 3 | |
2 | sbcrel 4444 | . . . 4 | |
3 | sbcal 2865 | . . . . 5 | |
4 | sbcal 2865 | . . . . . . 7 | |
5 | sbcal 2865 | . . . . . . . . 9 | |
6 | sbcimg 2855 | . . . . . . . . . . 11 | |
7 | sbcan 2856 | . . . . . . . . . . . . 13 | |
8 | sbcbrg 3834 | . . . . . . . . . . . . . . 15 | |
9 | csbconstg 2920 | . . . . . . . . . . . . . . . 16 | |
10 | csbconstg 2920 | . . . . . . . . . . . . . . . 16 | |
11 | 9, 10 | breq12d 3798 | . . . . . . . . . . . . . . 15 |
12 | 8, 11 | bitrd 186 | . . . . . . . . . . . . . 14 |
13 | sbcbrg 3834 | . . . . . . . . . . . . . . 15 | |
14 | csbconstg 2920 | . . . . . . . . . . . . . . . 16 | |
15 | 9, 14 | breq12d 3798 | . . . . . . . . . . . . . . 15 |
16 | 13, 15 | bitrd 186 | . . . . . . . . . . . . . 14 |
17 | 12, 16 | anbi12d 456 | . . . . . . . . . . . . 13 |
18 | 7, 17 | syl5bb 190 | . . . . . . . . . . . 12 |
19 | sbcg 2883 | . . . . . . . . . . . 12 | |
20 | 18, 19 | imbi12d 232 | . . . . . . . . . . 11 |
21 | 6, 20 | bitrd 186 | . . . . . . . . . 10 |
22 | 21 | albidv 1745 | . . . . . . . . 9 |
23 | 5, 22 | syl5bb 190 | . . . . . . . 8 |
24 | 23 | albidv 1745 | . . . . . . 7 |
25 | 4, 24 | syl5bb 190 | . . . . . 6 |
26 | 25 | albidv 1745 | . . . . 5 |
27 | 3, 26 | syl5bb 190 | . . . 4 |
28 | 2, 27 | anbi12d 456 | . . 3 |
29 | 1, 28 | syl5bb 190 | . 2 |
30 | dffun2 4932 | . . 3 | |
31 | 30 | sbcbii 2873 | . 2 |
32 | dffun2 4932 | . 2 | |
33 | 29, 31, 32 | 3bitr4g 221 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wal 1282 wcel 1433 wsbc 2815 csb 2908 class class class wbr 3785 wrel 4368 wfun 4916 |
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-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-v 2603 df-sbc 2816 df-csb 2909 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-id 4048 df-rel 4370 df-cnv 4371 df-co 4372 df-fun 4924 |
This theorem is referenced by: sbcfng 5064 |
Copyright terms: Public domain | W3C validator |