Theorem eliminable2a 32841

Description: A theorem used to prove the base case of the Eliminability Theorem (see section comment). (Contributed by BJ, 19-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
eliminable2a	|- ( x = { y | ph } <-> A. z ( z e. x <-> z e. { y | ph } ) )

Distinct variable groups:   x, z   y, z   ph, z

Allowed substitution hints:   ph( x, y)

Proof of Theorem eliminable2a

Step	Hyp	Ref	Expression
1		dfcleq 2616	1 |- ( x = { y | ph } <-> A. z ( z e. x <-> z e. { y | ph } ) )

Syntax hints:   <-> wb 196   A.wal 1481   = wceq 1483   e. wcel 1990   {cab 2608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-cleq 2615

This theorem is referenced by: (None)