Theorem eqab 2373

Description: One direction of eqabb 2374. (Contributed by Wolf Lammen, 13-Feb-2025.)

Assertion

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

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem eqab

Step	Hyp	Ref	Expression
1		abid1 2372	. 2 |- A = { x | x e. A }
2		abbi 2357	. 2 |- ( A. x ( x e. A <-> ph ) -> { x | x e. A } = { x | ph } )
3	1, 2	eqtrid 2283	1 |- ( A. x ( x e. A <-> ph ) -> A = { x | ph } )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1400   = wceq 1402   e. wcel 2209   {cab 2224

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234

This theorem is referenced by: (None)