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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 2946 | . . 3 | |
2 | sbcrel 4620 | . . . 4 | |
3 | sbcal 2955 | . . . . 5 | |
4 | sbcal 2955 | . . . . . . 7 | |
5 | sbcal 2955 | . . . . . . . . 9 | |
6 | sbcimg 2945 | . . . . . . . . . . 11 | |
7 | sbcan 2946 | . . . . . . . . . . . . 13 | |
8 | sbcbrg 3977 | . . . . . . . . . . . . . . 15 | |
9 | csbconstg 3011 | . . . . . . . . . . . . . . . 16 | |
10 | csbconstg 3011 | . . . . . . . . . . . . . . . 16 | |
11 | 9, 10 | breq12d 3937 | . . . . . . . . . . . . . . 15 |
12 | 8, 11 | bitrd 187 | . . . . . . . . . . . . . 14 |
13 | sbcbrg 3977 | . . . . . . . . . . . . . . 15 | |
14 | csbconstg 3011 | . . . . . . . . . . . . . . . 16 | |
15 | 9, 14 | breq12d 3937 | . . . . . . . . . . . . . . 15 |
16 | 13, 15 | bitrd 187 | . . . . . . . . . . . . . 14 |
17 | 12, 16 | anbi12d 464 | . . . . . . . . . . . . 13 |
18 | 7, 17 | syl5bb 191 | . . . . . . . . . . . 12 |
19 | sbcg 2973 | . . . . . . . . . . . 12 | |
20 | 18, 19 | imbi12d 233 | . . . . . . . . . . 11 |
21 | 6, 20 | bitrd 187 | . . . . . . . . . 10 |
22 | 21 | albidv 1796 | . . . . . . . . 9 |
23 | 5, 22 | syl5bb 191 | . . . . . . . 8 |
24 | 23 | albidv 1796 | . . . . . . 7 |
25 | 4, 24 | syl5bb 191 | . . . . . 6 |
26 | 25 | albidv 1796 | . . . . 5 |
27 | 3, 26 | syl5bb 191 | . . . 4 |
28 | 2, 27 | anbi12d 464 | . . 3 |
29 | 1, 28 | syl5bb 191 | . 2 |
30 | dffun2 5128 | . . 3 | |
31 | 30 | sbcbii 2963 | . 2 |
32 | dffun2 5128 | . 2 | |
33 | 29, 31, 32 | 3bitr4g 222 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wal 1329 wcel 1480 wsbc 2904 csb 2998 class class class wbr 3924 wrel 4539 wfun 5112 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-v 2683 df-sbc 2905 df-csb 2999 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-id 4210 df-rel 4541 df-cnv 4542 df-co 4543 df-fun 5120 |
This theorem is referenced by: sbcfng 5265 |
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