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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 2917 | . . 3 | |
2 | 1 | a1i 9 | . 2 |
3 | nfnfc1 2284 | . . 3 | |
4 | id 19 | . . . 4 | |
5 | nfcvd 2282 | . . . 4 | |
6 | 4, 5 | nfeld 2297 | . . 3 |
7 | sbcex 2917 | . . . 4 | |
8 | 7 | 2a1i 27 | . . 3 |
9 | 3, 6, 8 | rexlimd2 2547 | . 2 |
10 | sbcco 2930 | . . . 4 | |
11 | simpl 108 | . . . . 5 | |
12 | sbsbc 2913 | . . . . . . 7 | |
13 | nfcv 2281 | . . . . . . . . 9 | |
14 | nfs1v 1912 | . . . . . . . . 9 | |
15 | 13, 14 | nfrexxy 2472 | . . . . . . . 8 |
16 | sbequ12 1744 | . . . . . . . . 9 | |
17 | 16 | rexbidv 2438 | . . . . . . . 8 |
18 | 15, 17 | sbie 1764 | . . . . . . 7 |
19 | 12, 18 | bitr3i 185 | . . . . . 6 |
20 | nfcvd 2282 | . . . . . . . . . 10 | |
21 | 20, 4 | nfeqd 2296 | . . . . . . . . 9 |
22 | 3, 21 | nfan1 1543 | . . . . . . . 8 |
23 | dfsbcq2 2912 | . . . . . . . . 9 | |
24 | 23 | adantl 275 | . . . . . . . 8 |
25 | 22, 24 | rexbid 2436 | . . . . . . 7 |
26 | 25 | adantll 467 | . . . . . 6 |
27 | 19, 26 | syl5bb 191 | . . . . 5 |
28 | 11, 27 | sbcied 2945 | . . . 4 |
29 | 10, 28 | syl5bbr 193 | . . 3 |
30 | 29 | expcom 115 | . 2 |
31 | 2, 9, 30 | pm5.21ndd 694 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wsb 1735 wnfc 2268 wrex 2417 cvv 2686 wsbc 2909 |
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-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 |
This theorem is referenced by: sbcrex 2988 |
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