Theorem bnj873 33070

Description: Technical lemma for bnj69 33154. 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
bnj873.4	|- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
bnj873.7	|- ( ph' <-> [. g / f ]. ph )
bnj873.8	|- ( ps' <-> [. g / f ]. ps )

Assertion

Ref	Expression
bnj873	|- B = { g | E. n e. D ( g Fn n /\ ph' /\ ps' ) }

Distinct variable groups:   D, f, g   f, n, g   ph, g   ps, g

Allowed substitution hints:   ph( f, n)   ps( f, n)   B( f, g, n)   D( n)   ph'( f, g, n)   ps'( f, g, n)

Proof of Theorem bnj873

Step	Hyp	Ref	Expression
1		bnj873.4	. 2 |- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
2		nfv 1683	. . 3 |- F/ g E. n e. D ( f Fn n /\ ph /\ ps )
3		nfcv 2629	. . . 4 |- F/_ f D
4		nfv 1683	. . . . 5 |- F/ f g Fn n
5		bnj873.7	. . . . . 6 |- ( ph' <-> [. g / f ]. ph )
6		nfsbc1v 3351	. . . . . 6 |- F/ f [. g / f ]. ph
7	5, 6	nfxfr 1625	. . . . 5 |- F/ f ph'
8		bnj873.8	. . . . . 6 |- ( ps' <-> [. g / f ]. ps )
9		nfsbc1v 3351	. . . . . 6 |- F/ f [. g / f ]. ps
10	8, 9	nfxfr 1625	. . . . 5 |- F/ f ps'
11	4, 7, 10	nf3an 1877	. . . 4 |- F/ f ( g Fn n /\ ph' /\ ps' )
12	3, 11	nfrex 2927	. . 3 |- F/ f E. n e. D ( g Fn n /\ ph' /\ ps' )
13		fneq1 5668	. . . . 5 |- ( f = g -> ( f Fn n <-> g Fn n ) )
14		sbceq1a 3342	. . . . . 6 |- ( f = g -> ( ph <-> [. g / f ]. ph ) )
15	14, 5	syl6bbr 263	. . . . 5 |- ( f = g -> ( ph <-> ph' ) )
16		sbceq1a 3342	. . . . . 6 |- ( f = g -> ( ps <-> [. g / f ]. ps ) )
17	16, 8	syl6bbr 263	. . . . 5 |- ( f = g -> ( ps <-> ps' ) )
18	13, 15, 17	3anbi123d 1299	. . . 4 |- ( f = g -> ( ( f Fn n /\ ph /\ ps ) <-> ( g Fn n /\ ph' /\ ps' ) ) )
19	18	rexbidv 2973	. . 3 |- ( f = g -> ( E. n e. D ( f Fn n /\ ph /\ ps ) <-> E. n e. D ( g Fn n /\ ph' /\ ps' ) ) )
20	2, 12, 19	cbvab 2608	. 2 |- { f | E. n e. D ( f Fn n /\ ph /\ ps ) } = { g | E. n e. D ( g Fn n /\ ph' /\ ps' ) }
21	1, 20	eqtri 2496	1 |- B = { g | E. n e. D ( g Fn n /\ ph' /\ ps' ) }

Syntax hints:   <-> wb 184   /\ w3a 973   = wceq 1379   {cab 2452   E.wrex 2815   [.wsbc 3331   Fn wfn 5582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-fun 5589  df-fn 5590

This theorem is referenced by:  bnj849  33071  bnj893  33074