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Theorem bnj1138 30113
 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 2680 . 2 (𝑋𝐴𝑋 ∈ (𝐵𝐶))
3 elun 3715 . 2 (𝑋 ∈ (𝐵𝐶) ↔ (𝑋𝐵𝑋𝐶))
42, 3bitri 263 1 (𝑋𝐴 ↔ (𝑋𝐵𝑋𝐶))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∨ wo 382   = wceq 1475   ∈ wcel 1977   ∪ cun 3538 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545 This theorem is referenced by:  bnj1424  30163  bnj1408  30358  bnj1417  30363
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