Theorem bnj984 31022

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
bnj984.3	|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
bnj984.11	|- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }

Assertion

Ref	Expression
bnj984	|- ( G e. A -> ( G e. B <-> [. G / f ]. E. n ch ) )

Proof of Theorem bnj984

Step	Hyp	Ref	Expression
1		sbc8g 3443	. . 3 |- ( G e. A -> ( [. G / f ]. E. n e. D ( f Fn n /\ ph /\ ps ) <-> G e. { f | E. n e. D ( f Fn n /\ ph /\ ps ) } ) )
2		bnj984.11	. . . 4 |- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
3	2	eleq2i 2693	. . 3 |- ( G e. B <-> G e. { f | E. n e. D ( f Fn n /\ ph /\ ps ) } )
4	1, 3	syl6rbbr 279	. 2 |- ( G e. A -> ( G e. B <-> [. G / f ]. E. n e. D ( f Fn n /\ ph /\ ps ) ) )
5		df-rex 2918	. . . 4 |- ( E. n e. D ( f Fn n /\ ph /\ ps ) <-> E. n ( n e. D /\ ( f Fn n /\ ph /\ ps ) ) )
6		bnj984.3	. . . . 5 |- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
7		bnj252 30769	. . . . 5 |- ( ( n e. D /\ f Fn n /\ ph /\ ps ) <-> ( n e. D /\ ( f Fn n /\ ph /\ ps ) ) )
8	6, 7	bitri 264	. . . 4 |- ( ch <-> ( n e. D /\ ( f Fn n /\ ph /\ ps ) ) )
9	5, 8	bnj133 30793	. . 3 |- ( E. n e. D ( f Fn n /\ ph /\ ps ) <-> E. n ch )
10	9	sbcbii 3491	. 2 |- ( [. G / f ]. E. n e. D ( f Fn n /\ ph /\ ps ) <-> [. G / f ]. E. n ch )
11	4, 10	syl6bb 276	1 |- ( G e. A -> ( G e. B <-> [. G / f ]. E. n ch ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   E.wrex 2913   [.wsbc 3435   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-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-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-rex 2918  df-v 3202  df-sbc 3436  df-bnj17 30753

This theorem is referenced by:  bnj985  31023