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Theorem bnj1138 32085
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 2907 . 2 (𝑋𝐴𝑋 ∈ (𝐵𝐶))
3 elun 4110 . 2 (𝑋 ∈ (𝐵𝐶) ↔ (𝑋𝐵𝑋𝐶))
42, 3bitri 278 1 (𝑋𝐴 ↔ (𝑋𝐵𝑋𝐶))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wo 844   = wceq 1538  wcel 2115  cun 3917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3482  df-un 3924
This theorem is referenced by:  bnj1424  32135  bnj1408  32333  bnj1417  32338
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