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Theorem bnj1149 32089
 Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1149.1 (𝜑𝐴 ∈ V)
bnj1149.2 (𝜑𝐵 ∈ V)
Assertion
Ref Expression
bnj1149 (𝜑 → (𝐴𝐵) ∈ V)

Proof of Theorem bnj1149
StepHypRef Expression
1 bnj1149.1 . 2 (𝜑𝐴 ∈ V)
2 bnj1149.2 . 2 (𝜑𝐵 ∈ V)
3 unexg 7462 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵) ∈ V)
41, 2, 3syl2anc 587 1 (𝜑 → (𝐴𝐵) ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2115  Vcvv 3480   ∪ 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-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317  ax-un 7451 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-sn 4550  df-pr 4552  df-uni 4825 This theorem is referenced by:  bnj1136  32294  bnj1413  32332  bnj1452  32349  bnj1489  32353
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