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Mirrors > Home > ILE Home > Th. List > eqrdav | Unicode version |
Description: Deduce equality of classes from an equivalence of membership that depends on the membership variable. (Contributed by NM, 7-Nov-2008.) |
Ref | Expression |
---|---|
eqrdav.1 | |
eqrdav.2 | |
eqrdav.3 |
Ref | Expression |
---|---|
eqrdav |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqrdav.1 | . . . 4 | |
2 | eqrdav.3 | . . . . . 6 | |
3 | 2 | biimpd 132 | . . . . 5 |
4 | 3 | impancom 247 | . . . 4 |
5 | 1, 4 | mpd 13 | . . 3 |
6 | eqrdav.2 | . . . 4 | |
7 | 2 | exbiri 364 | . . . . . 6 |
8 | 7 | com23 72 | . . . . 5 |
9 | 8 | imp 115 | . . . 4 |
10 | 6, 9 | mpd 13 | . . 3 |
11 | 5, 10 | impbida 528 | . 2 |
12 | 11 | eqrdv 2038 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wcel 1393 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-gen 1338 ax-17 1419 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-cleq 2033 |
This theorem is referenced by: fzdifsuc 8943 |
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