Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > sbciegft | Structured version Visualization version Unicode version |
Description: Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 3467.) (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Ref | Expression |
---|---|
sbciegft |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbc5 3460 | . . 3 | |
2 | biimp 205 | . . . . . . . 8 | |
3 | 2 | imim2i 16 | . . . . . . 7 |
4 | 3 | impd 447 | . . . . . 6 |
5 | 4 | alimi 1739 | . . . . 5 |
6 | 19.23t 2079 | . . . . . 6 | |
7 | 6 | biimpa 501 | . . . . 5 |
8 | 5, 7 | sylan2 491 | . . . 4 |
9 | 8 | 3adant1 1079 | . . 3 |
10 | 1, 9 | syl5bi 232 | . 2 |
11 | biimpr 210 | . . . . . . . 8 | |
12 | 11 | imim2i 16 | . . . . . . 7 |
13 | 12 | com23 86 | . . . . . 6 |
14 | 13 | alimi 1739 | . . . . 5 |
15 | 19.21t 2073 | . . . . . 6 | |
16 | 15 | biimpa 501 | . . . . 5 |
17 | 14, 16 | sylan2 491 | . . . 4 |
18 | 17 | 3adant1 1079 | . . 3 |
19 | sbc6g 3461 | . . . 4 | |
20 | 19 | 3ad2ant1 1082 | . . 3 |
21 | 18, 20 | sylibrd 249 | . 2 |
22 | 10, 21 | impbid 202 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wal 1481 wceq 1483 wex 1704 wnf 1708 wcel 1990 wsbc 3435 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-v 3202 df-sbc 3436 |
This theorem is referenced by: sbciegf 3467 sbciedf 3471 |
Copyright terms: Public domain | W3C validator |