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Theorem bnj1212 31970
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 4054 . 2 𝐵𝐴
3 bnj1212.2 . . 3 (𝜃 ↔ (𝜒𝑥𝐵𝜏))
43simp2bi 1138 . 2 (𝜃𝑥𝐵)
52, 4bnj1213 31969 1 (𝜃𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  w3a 1079   = wceq 1528  wcel 2105  {crab 3139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-in 3940  df-ss 3949
This theorem is referenced by:  bnj1204  32181  bnj1296  32190  bnj1415  32207  bnj1421  32211  bnj1442  32218  bnj1452  32221  bnj1489  32225
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