Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > raaan2 | Structured version Visualization version Unicode version |
Description: Rearrange restricted quantifiers with two different restricting classes, analogous to raaan 4082. It is necessary that either both restricting classes are empty or both are not empty. (Contributed by Alexander van der Vekens, 29-Jun-2017.) |
Ref | Expression |
---|---|
raaan2.1 | |
raaan2.2 |
Ref | Expression |
---|---|
raaan2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfbi3 994 | . 2 | |
2 | rzal 4073 | . . . . 5 | |
3 | 2 | adantr 481 | . . . 4 |
4 | rzal 4073 | . . . . 5 | |
5 | 4 | adantr 481 | . . . 4 |
6 | rzal 4073 | . . . . 5 | |
7 | 6 | adantl 482 | . . . 4 |
8 | pm5.1 902 | . . . 4 | |
9 | 3, 5, 7, 8 | syl12anc 1324 | . . 3 |
10 | df-ne 2795 | . . . . 5 | |
11 | raaan2.1 | . . . . . . 7 | |
12 | 11 | r19.28z 4063 | . . . . . 6 |
13 | 12 | ralbidv 2986 | . . . . 5 |
14 | 10, 13 | sylbir 225 | . . . 4 |
15 | df-ne 2795 | . . . . 5 | |
16 | nfcv 2764 | . . . . . . 7 | |
17 | raaan2.2 | . . . . . . 7 | |
18 | 16, 17 | nfral 2945 | . . . . . 6 |
19 | 18 | r19.27z 4070 | . . . . 5 |
20 | 15, 19 | sylbir 225 | . . . 4 |
21 | 14, 20 | sylan9bbr 737 | . . 3 |
22 | 9, 21 | jaoi 394 | . 2 |
23 | 1, 22 | sylbi 207 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wo 383 wa 384 wceq 1483 wnf 1708 wne 2794 wral 2912 c0 3915 |
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-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-v 3202 df-dif 3577 df-nul 3916 |
This theorem is referenced by: 2reu4a 41189 |
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