Theorem bnj1538 30925

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj1538.1	|- A = { x e. B | ph }

Assertion

Ref	Expression
bnj1538	|- ( x e. A -> ph )

Proof of Theorem bnj1538

Step	Hyp	Ref	Expression
1		bnj1538.1	. . 3 |- A = { x e. B | ph }
2	1	rabeq2i 3197	. 2 |- ( x e. A <-> ( x e. B /\ ph ) )
3	2	simprbi 480	1 |- ( x e. A -> ph )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {crab 2916

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-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-tru 1486  df-ex 1705  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-rab 2921

This theorem is referenced by:  bnj1279  31086  bnj1311  31092  bnj1418  31108  bnj1312  31126  bnj1523  31139