Theorem bnj985 31023

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
bnj985.3	|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
bnj985.6	|- ( ch' <-> [. p / n ]. ch )
bnj985.9	|- ( ch" <-> [. G / f ]. ch' )
bnj985.11	|- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
bnj985.13	|- G = ( f u. { <. n , C >. } )

Assertion

Ref	Expression
bnj985	|- ( G e. B <-> E. p ch" )

Distinct variable groups:   G, p   ch, p   f, p

Allowed substitution hints:   ph( f, n, p)   ps( f, n, p)   ch( f, n)   B( f, n, p)   C( f, n, p)   D( f, n, p)   G( f, n)   ch'( f, n, p)   ch"( f, n, p)

Proof of Theorem bnj985

Step	Hyp	Ref	Expression
1		bnj985.13	. . . 4 |- G = ( f u. { <. n , C >. } )
2	1	bnj918 30836	. . 3 |- G e. _V
3		bnj985.3	. . . 4 |- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
4		bnj985.11	. . . 4 |- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
5	3, 4	bnj984 31022	. . 3 |- ( G e. _V -> ( G e. B <-> [. G / f ]. E. n ch ) )
6	2, 5	ax-mp 5	. 2 |- ( G e. B <-> [. G / f ]. E. n ch )
7		sbcex2 3486	. . 3 |- ( [. G / f ]. E. p ch' <-> E. p [. G / f ]. ch' )
8		nfv 1843	. . . . . . 7 |- F/ p ch
9	8	sb8e 2425	. . . . . 6 |- ( E. n ch <-> E. p [ p / n ] ch )
10		sbsbc 3439	. . . . . . 7 |- ( [ p / n ] ch <-> [. p / n ]. ch )
11	10	exbii 1774	. . . . . 6 |- ( E. p [ p / n ] ch <-> E. p [. p / n ]. ch )
12	9, 11	bitri 264	. . . . 5 |- ( E. n ch <-> E. p [. p / n ]. ch )
13		bnj985.6	. . . . 5 |- ( ch' <-> [. p / n ]. ch )
14	12, 13	bnj133 30793	. . . 4 |- ( E. n ch <-> E. p ch' )
15	14	sbcbii 3491	. . 3 |- ( [. G / f ]. E. n ch <-> [. G / f ]. E. p ch' )
16		bnj985.9	. . . 4 |- ( ch" <-> [. G / f ]. ch' )
17	16	exbii 1774	. . 3 |- ( E. p ch" <-> E. p [. G / f ]. ch' )
18	7, 15, 17	3bitr4i 292	. 2 |- ( [. G / f ]. E. n ch <-> E. p ch" )
19	6, 18	bitri 264	1 |- ( G e. B <-> E. p ch" )

Syntax hints:   <-> wb 196   /\ w3a 1037   = wceq 1483   E.wex 1704   [wsb 1880   e. wcel 1990   {cab 2608   E.wrex 2913   _Vcvv 3200   [.wsbc 3435   u. cun 3572   {csn 4177   <.cop 4183   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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

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-nfc 2753  df-rex 2918  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-nul 3916  df-sn 4178  df-pr 4180  df-uni 4437  df-bnj17 30753

This theorem is referenced by:  bnj1018  31032