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Theorem bnj1138 31959
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj1138.1 𝐴 = (𝐵𝐶)
Assertion
Ref Expression
bnj1138 (𝑋𝐴 ↔ (𝑋𝐵𝑋𝐶))

Proof of Theorem bnj1138
StepHypRef Expression
1 bnj1138.1 . . 3 𝐴 = (𝐵𝐶)
21eleq2i 2901 . 2 (𝑋𝐴𝑋 ∈ (𝐵𝐶))
3 elun 4122 . 2 (𝑋 ∈ (𝐵𝐶) ↔ (𝑋𝐵𝑋𝐶))
42, 3bitri 276 1 (𝑋𝐴 ↔ (𝑋𝐵𝑋𝐶))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wo 841   = wceq 1528  wcel 2105  cun 3931
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494  df-un 3938
This theorem is referenced by:  bnj1424  32009  bnj1408  32205  bnj1417  32210
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