Theorem opabidw 4380

Description: The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Version of opabid 4379 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 4379	1 |- ( <. x , y >. e. { <. x , y >. | ph } <-> ph )

Syntax hints:   <-> wb 105   e. wcel 2205   <.cop 3697   {copab 4175

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 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-opab 4177

This theorem is referenced by:  lgsquadlem1  16062  lgsquadlem2  16063