Theorem opabidw 4344

Description: The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Version of opabid 4343 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 4343	1 |- ( <. x , y >. e. { <. x , y >. | ph } <-> ph )

Syntax hints:   <-> wb 105   e. wcel 2200   <.cop 3669   {copab 4143

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-opab 4145

This theorem is referenced by:  lgsquadlem1  15741  lgsquadlem2  15742