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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 |
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eqrdav.2 |
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eqrdav.3 |
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Ref | Expression |
---|---|
eqrdav |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqrdav.1 |
. . . 4
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2 | eqrdav.3 |
. . . . . 6
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3 | 2 | biimpd 143 |
. . . . 5
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4 | 3 | impancom 258 |
. . . 4
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5 | 1, 4 | mpd 13 |
. . 3
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6 | eqrdav.2 |
. . . 4
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7 | 2 | exbiri 380 |
. . . . . 6
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8 | 7 | com23 78 |
. . . . 5
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9 | 8 | imp 123 |
. . . 4
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10 | 6, 9 | mpd 13 |
. . 3
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11 | 5, 10 | impbida 586 |
. 2
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12 | 11 | eqrdv 2138 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() |
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-5 1424 ax-gen 1426 ax-17 1507 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-cleq 2133 |
This theorem is referenced by: supminfex 9419 fzdifsuc 9892 |
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