Theorem bnj1230 30873

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

Hypothesis

Ref	Expression
bnj1230.1	|- B = { x e. A | ph }

Assertion

Ref	Expression
bnj1230	|- ( y e. B -> A. x y e. B )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)   A( x, y)   B( x, y)

Proof of Theorem bnj1230

Step	Hyp	Ref	Expression
1		bnj1230.1	. . 3 |- B = { x e. A | ph }
2		nfrab1 3122	. . 3 |- F/_ x { x e. A | ph }
3	1, 2	nfcxfr 2762	. 2 |- F/_ x B
4	3	nfcrii 2757	1 |- ( y e. B -> A. x y e. B )

Syntax hints:   -> wi 4   A.wal 1481   = 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-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921

This theorem is referenced by:  bnj1312  31126