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Theorem bnj1138 31214
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 2845 . 2 (𝑋𝐴𝑋 ∈ (𝐵𝐶))
3 elun 3911 . 2 (𝑋 ∈ (𝐵𝐶) ↔ (𝑋𝐵𝑋𝐶))
42, 3bitri 265 1 (𝑋𝐴 ↔ (𝑋𝐵𝑋𝐶))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wo 863   = wceq 1634  wcel 2148  cun 3727
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-v 3357  df-un 3734
This theorem is referenced by:  bnj1424  31264  bnj1408  31459  bnj1417  31464
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