Theorem opabidw 4307

Description: The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Version of opabid 4306 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 4306	1 |- ( <. x , y >. e. { <. x , y >. | ph } <-> ph )

Syntax hints:   <-> wb 105   e. wcel 2177   <.cop 3637   {copab 4108

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-pow 4222  ax-pr 4257

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3171  df-in 3173  df-ss 3180  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-opab 4110

This theorem is referenced by:  lgsquadlem1  15598  lgsquadlem2  15599