Theorem bnj873 30994

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
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 1843	. . 3 |- F/ g E. n e. D ( f Fn n /\ ph /\ ps )
3		nfcv 2764	. . . 4 |- F/_ f D
4		nfv 1843	. . . . 5 |- F/ f g Fn n
5		bnj873.7	. . . . . 6 |- ( ph' <-> [. g / f ]. ph )
6		nfsbc1v 3455	. . . . . 6 |- F/ f [. g / f ]. ph
7	5, 6	nfxfr 1779	. . . . 5 |- F/ f ph'
8		bnj873.8	. . . . . 6 |- ( ps' <-> [. g / f ]. ps )
9		nfsbc1v 3455	. . . . . 6 |- F/ f [. g / f ]. ps
10	8, 9	nfxfr 1779	. . . . 5 |- F/ f ps'
11	4, 7, 10	nf3an 1831	. . . 4 |- F/ f ( g Fn n /\ ph' /\ ps' )
12	3, 11	nfrex 3007	. . 3 |- F/ f E. n e. D ( g Fn n /\ ph' /\ ps' )
13		fneq1 5979	. . . . 5 |- ( f = g -> ( f Fn n <-> g Fn n ) )
14		sbceq1a 3446	. . . . . 6 |- ( f = g -> ( ph <-> [. g / f ]. ph ) )
15	14, 5	syl6bbr 278	. . . . 5 |- ( f = g -> ( ph <-> ph' ) )
16		sbceq1a 3446	. . . . . 6 |- ( f = g -> ( ps <-> [. g / f ]. ps ) )
17	16, 8	syl6bbr 278	. . . . 5 |- ( f = g -> ( ps <-> ps' ) )
18	13, 15, 17	3anbi123d 1399	. . . 4 |- ( f = g -> ( ( f Fn n /\ ph /\ ps ) <-> ( g Fn n /\ ph' /\ ps' ) ) )
19	18	rexbidv 3052	. . 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 2746	. 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 2644	1 |- B = { g | E. n e. D ( g Fn n /\ ph' /\ ps' ) }

Syntax hints:   <-> wb 196   /\ w3a 1037   = wceq 1483   {cab 2608   E.wrex 2913   [.wsbc 3435   Fn wfn 5883

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-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-fun 5890  df-fn 5891

This theorem is referenced by:  bnj849  30995  bnj893  30998