Theorem opabidw 4374

Description: The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Version of opabid 4373 with a disjoint variable condition. (Contributed by NM, 14-Apr-1995.) (Revised by GG, 26-Jan-2024.)

Assertion

Ref	Expression
opabidw	|- ( <. x , y >. e. { <. x , y >. | ph } <-> ph )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem opabidw

Step	Hyp	Ref	Expression
1		opabid 4373	1 |- ( <. x , y >. e. { <. x , y >. | ph } <-> ph )

Syntax hints:   <-> wb 105   e. wcel 2203   <.cop 3691   {copab 4169

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2814  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-opab 4171

This theorem is referenced by:  lgsquadlem1  15942  lgsquadlem2  15943