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Mirrors > Home > ILE Home > Th. List > sbcrext | Unicode version |
Description: Interchange class substitution and restricted existential quantifier. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) |
Ref | Expression |
---|---|
sbcrext |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcex 2823 | . . 3 | |
2 | 1 | a1i 9 | . 2 |
3 | nfnfc1 2222 | . . 3 | |
4 | id 19 | . . . 4 | |
5 | nfcvd 2220 | . . . 4 | |
6 | 4, 5 | nfeld 2234 | . . 3 |
7 | sbcex 2823 | . . . 4 | |
8 | 7 | 2a1i 27 | . . 3 |
9 | 3, 6, 8 | rexlimd2 2475 | . 2 |
10 | sbcco 2836 | . . . 4 | |
11 | simpl 107 | . . . . 5 | |
12 | sbsbc 2819 | . . . . . . 7 | |
13 | nfcv 2219 | . . . . . . . . 9 | |
14 | nfs1v 1856 | . . . . . . . . 9 | |
15 | 13, 14 | nfrexxy 2403 | . . . . . . . 8 |
16 | sbequ12 1694 | . . . . . . . . 9 | |
17 | 16 | rexbidv 2369 | . . . . . . . 8 |
18 | 15, 17 | sbie 1714 | . . . . . . 7 |
19 | 12, 18 | bitr3i 184 | . . . . . 6 |
20 | nfcvd 2220 | . . . . . . . . . 10 | |
21 | 20, 4 | nfeqd 2233 | . . . . . . . . 9 |
22 | 3, 21 | nfan1 1496 | . . . . . . . 8 |
23 | dfsbcq2 2818 | . . . . . . . . 9 | |
24 | 23 | adantl 271 | . . . . . . . 8 |
25 | 22, 24 | rexbid 2367 | . . . . . . 7 |
26 | 25 | adantll 459 | . . . . . 6 |
27 | 19, 26 | syl5bb 190 | . . . . 5 |
28 | 11, 27 | sbcied 2850 | . . . 4 |
29 | 10, 28 | syl5bbr 192 | . . 3 |
30 | 29 | expcom 114 | . 2 |
31 | 2, 9, 30 | pm5.21ndd 653 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wceq 1284 wcel 1433 wsb 1685 wnfc 2206 wrex 2349 cvv 2601 wsbc 2815 |
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-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-sbc 2816 |
This theorem is referenced by: sbcrex 2893 |
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