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Theorem bnj1212 34813
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1212.1 𝐵 = {𝑥𝐴𝜑}
bnj1212.2 (𝜃 ↔ (𝜒𝑥𝐵𝜏))
Assertion
Ref Expression
bnj1212 (𝜃𝑥𝐴)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜒(𝑥)   𝜃(𝑥)   𝜏(𝑥)   𝐵(𝑥)

Proof of Theorem bnj1212
StepHypRef Expression
1 bnj1212.1 . . 3 𝐵 = {𝑥𝐴𝜑}
21ssrab3 4082 . 2 𝐵𝐴
3 bnj1212.2 . . 3 (𝜃 ↔ (𝜒𝑥𝐵𝜏))
43simp2bi 1147 . 2 (𝜃𝑥𝐵)
52, 4bnj1213 34812 1 (𝜃𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1087   = wceq 1540  wcel 2108  {crab 3436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-ss 3968
This theorem is referenced by:  bnj1204  35026  bnj1296  35035  bnj1415  35052  bnj1421  35056  bnj1442  35063  bnj1452  35066  bnj1489  35070
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