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Theorem prexgOLD 3937
 Description: The Axiom of Pairing using class variables. Theorem 7.13 of [Quine] p. 51, but restricted to classes which exist. For proper classes, see prprc 3471, prprc1 3469, and prprc2 3470. This is a special case of prexg 3938 and new proofs should use prexg 3938 instead. (Contributed by Jim Kingdon, 25-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.) TODO: replace its uses by uses of prexg 3938 and then remove it.
Assertion
Ref Expression
prexgOLD ((A V B V) → {A, B} V)

Proof of Theorem prexgOLD
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 preq2 3439 . . . . . 6 (y = B → {x, y} = {x, B})
21eleq1d 2103 . . . . 5 (y = B → ({x, y} V ↔ {x, B} V))
3 zfpair2 3936 . . . . 5 {x, y} V
42, 3vtoclg 2607 . . . 4 (B V → {x, B} V)
5 preq1 3438 . . . . 5 (x = A → {x, B} = {A, B})
65eleq1d 2103 . . . 4 (x = A → ({x, B} V ↔ {A, B} V))
74, 6syl5ib 143 . . 3 (x = A → (B V → {A, B} V))
87vtocleg 2618 . 2 (A V → (B V → {A, B} V))
98imp 115 1 ((A V B V) → {A, B} V)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∈ wcel 1390  Vcvv 2551  {cpr 3368 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374 This theorem is referenced by:  prelpwi  3941  opexgOLD  3956  opi2  3961  opth  3965  opeqsn  3980  opeqpr  3981  uniop  3983  unex  4142  op1stb  4175  op1stbg  4176  opthreg  4234  relop  4429
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