Theorem bnj873 29731

Description: Technical lemma for bnj69 29815. 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 1751	. . 3 |- F/ g E. n e. D ( f Fn n /\ ph /\ ps )
3		nfcv 2584	. . . 4 |- F/_ f D
4		nfv 1751	. . . . 5 |- F/ f g Fn n
5		bnj873.7	. . . . . 6 |- ( ph' <-> [. g / f ]. ph )
6		nfsbc1v 3319	. . . . . 6 |- F/ f [. g / f ]. ph
7	5, 6	nfxfr 1692	. . . . 5 |- F/ f ph'
8		bnj873.8	. . . . . 6 |- ( ps' <-> [. g / f ]. ps )
9		nfsbc1v 3319	. . . . . 6 |- F/ f [. g / f ]. ps
10	8, 9	nfxfr 1692	. . . . 5 |- F/ f ps'
11	4, 7, 10	nf3an 1986	. . . 4 |- F/ f ( g Fn n /\ ph' /\ ps' )
12	3, 11	nfrex 2888	. . 3 |- F/ f E. n e. D ( g Fn n /\ ph' /\ ps' )
13		fneq1 5679	. . . . 5 |- ( f = g -> ( f Fn n <-> g Fn n ) )
14		sbceq1a 3310	. . . . . 6 |- ( f = g -> ( ph <-> [. g / f ]. ph ) )
15	14, 5	syl6bbr 266	. . . . 5 |- ( f = g -> ( ph <-> ph' ) )
16		sbceq1a 3310	. . . . . 6 |- ( f = g -> ( ps <-> [. g / f ]. ps ) )
17	16, 8	syl6bbr 266	. . . . 5 |- ( f = g -> ( ps <-> ps' ) )
18	13, 15, 17	3anbi123d 1335	. . . 4 |- ( f = g -> ( ( f Fn n /\ ph /\ ps ) <-> ( g Fn n /\ ph' /\ ps' ) ) )
19	18	rexbidv 2939	. . 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 2563	. 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 2451	1 |- B = { g | E. n e. D ( g Fn n /\ ph' /\ ps' ) }

Syntax hints:   <-> wb 187   /\ w3a 982   = wceq 1437   {cab 2407   E.wrex 2776   [.wsbc 3299   Fn wfn 5593

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-br 4421  df-opab 4480  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-fun 5600  df-fn 5601

This theorem is referenced by:  bnj849  29732  bnj893  29735