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Theorem ineqri 3343
Description: Inference from membership to intersection. (Contributed by NM, 5-Aug-1993.)
Hypothesis
Ref Expression
ineqri.1 ((𝑥𝐴𝑥𝐵) ↔ 𝑥𝐶)
Assertion
Ref Expression
ineqri (𝐴𝐵) = 𝐶
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem ineqri
StepHypRef Expression
1 elin 3333 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
2 ineqri.1 . . 3 ((𝑥𝐴𝑥𝐵) ↔ 𝑥𝐶)
31, 2bitri 184 . 2 (𝑥 ∈ (𝐴𝐵) ↔ 𝑥𝐶)
43eqriv 2186 1 (𝐴𝐵) = 𝐶
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105   = wceq 1364  wcel 2160  cin 3143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150
This theorem is referenced by:  inidm  3359  inass  3360  indi  3397  inab  3418  in0  3472  pwin  4300  dmres  4946
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