Mathbox for Jonathan Ben-Naim < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1212 Structured version   Visualization version   GIF version

Theorem bnj1212 30124
 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}
21bnj21 30037 . 2 𝐵𝐴
3 bnj1212.2 . . 3 (𝜃 ↔ (𝜒𝑥𝐵𝜏))
43simp2bi 1070 . 2 (𝜃𝑥𝐵)
52, 4bnj1213 30123 1 (𝜃𝑥𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {crab 2900 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-3an 1033  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-rab 2905  df-in 3547  df-ss 3554 This theorem is referenced by:  bnj1204  30334  bnj1296  30343  bnj1415  30360  bnj1421  30364  bnj1442  30371  bnj1452  30374  bnj1489  30378
 Copyright terms: Public domain W3C validator