Theorem bnj1083 31046

Description: Technical lemma for bnj69 31078. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1083.3	|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
bnj1083.8	|- K = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }

Assertion

Ref	Expression
bnj1083	|- ( f e. K <-> E. n ch )

Proof of Theorem bnj1083

Step	Hyp	Ref	Expression
1		df-rex 2918	. 2 |- ( E. n e. D ( f Fn n /\ ph /\ ps ) <-> E. n ( n e. D /\ ( f Fn n /\ ph /\ ps ) ) )
2		bnj1083.8	. . 3 |- K = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
3	2	abeq2i 2735	. 2 |- ( f e. K <-> E. n e. D ( f Fn n /\ ph /\ ps ) )
4		bnj1083.3	. . . 4 |- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
5		bnj252 30769	. . . 4 |- ( ( n e. D /\ f Fn n /\ ph /\ ps ) <-> ( n e. D /\ ( f Fn n /\ ph /\ ps ) ) )
6	4, 5	bitri 264	. . 3 |- ( ch <-> ( n e. D /\ ( f Fn n /\ ph /\ ps ) ) )
7	6	exbii 1774	. 2 |- ( E. n ch <-> E. n ( n e. D /\ ( f Fn n /\ ph /\ ps ) ) )
8	1, 3, 7	3bitr4i 292	1 |- ( f e. K <-> E. n ch )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   E.wrex 2913   Fn wfn 5883   /\ w-bnj17 30752

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-3an 1039  df-tru 1486  df-ex 1705  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-rex 2918  df-bnj17 30753

This theorem is referenced by:  bnj1121  31053  bnj1145  31061