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Theorem bnj1138 30985
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 2722 . 2 (𝑋𝐴𝑋 ∈ (𝐵𝐶))
3 elun 3786 . 2 (𝑋 ∈ (𝐵𝐶) ↔ (𝑋𝐵𝑋𝐶))
42, 3bitri 264 1 (𝑋𝐴 ↔ (𝑋𝐵𝑋𝐶))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wo 382   = wceq 1523  wcel 2030  cun 3605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-un 3612
This theorem is referenced by:  bnj1424  31035  bnj1408  31230  bnj1417  31235
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