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Theorem opelopabg 3975
Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 19-Dec-2013.)
Hypotheses
Ref Expression
opelopabg.1 (x = A → (φψ))
opelopabg.2 (y = B → (ψχ))
Assertion
Ref Expression
opelopabg ((A 𝑉 B 𝑊) → (⟨A, B {⟨x, y⟩ ∣ φ} ↔ χ))
Distinct variable groups:   x,y,A   x,B,y   χ,x,y
Allowed substitution hints:   φ(x,y)   ψ(x,y)   𝑉(x,y)   𝑊(x,y)

Proof of Theorem opelopabg
StepHypRef Expression
1 opelopabg.1 . . 3 (x = A → (φψ))
2 opelopabg.2 . . 3 (y = B → (ψχ))
31, 2sylan9bb 438 . 2 ((x = A y = B) → (φχ))
43opelopabga 3970 1 ((A 𝑉 B 𝑊) → (⟨A, B {⟨x, y⟩ ∣ φ} ↔ χ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1226   wcel 1370  cop 3349  {copab 3787
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-opab 3789
This theorem is referenced by:  opelopab  3978  fvopab3g  5166  fvopab3ig  5167  f1oiso  5386  ov  5539  ovg  5558  elopabi  5740  elinp  6322
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