Theorem bnj919 29650

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj919.1	|- ( ch <-> ( n e. D /\ F Fn n /\ ph /\ ps ) )
bnj919.2	|- ( ph' <-> [. P / n ]. ph )
bnj919.3	|- ( ps' <-> [. P / n ]. ps )
bnj919.4	|- ( ch' <-> [. P / n ]. ch )
bnj919.5	|- P e. _V

Assertion

Ref	Expression
bnj919	|- ( ch' <-> ( P e. D /\ F Fn P /\ ph' /\ ps' ) )

Distinct variable groups:   D, n   n, F   P, n

Allowed substitution hints:   ph( n)   ps( n)   ch( n)   ph'( n)   ps'( n)   ch'( n)

Proof of Theorem bnj919

Step	Hyp	Ref	Expression
1		bnj919.4	. 2 |- ( ch' <-> [. P / n ]. ch )
2		bnj919.1	. . 3 |- ( ch <-> ( n e. D /\ F Fn n /\ ph /\ ps ) )
3	2	sbcbii 3311	. 2 |- ( [. P / n ]. ch <-> [. P / n ]. ( n e. D /\ F Fn n /\ ph /\ ps ) )
4		bnj919.5	. . 3 |- P e. _V
5		df-bnj17 29564	. . . . 5 |- ( ( P e. D /\ F Fn P /\ ph' /\ ps' ) <-> ( ( P e. D /\ F Fn P /\ ph' ) /\ ps' ) )
6		nfv 1769	. . . . . . 7 |- F/ n P e. D
7		nfv 1769	. . . . . . 7 |- F/ n F Fn P
8		bnj919.2	. . . . . . . 8 |- ( ph' <-> [. P / n ]. ph )
9		nfsbc1v 3275	. . . . . . . 8 |- F/ n [. P / n ]. ph
10	8, 9	nfxfr 1704	. . . . . . 7 |- F/ n ph'
11	6, 7, 10	nf3an 2033	. . . . . 6 |- F/ n ( P e. D /\ F Fn P /\ ph' )
12		bnj919.3	. . . . . . 7 |- ( ps' <-> [. P / n ]. ps )
13		nfsbc1v 3275	. . . . . . 7 |- F/ n [. P / n ]. ps
14	12, 13	nfxfr 1704	. . . . . 6 |- F/ n ps'
15	11, 14	nfan 2031	. . . . 5 |- F/ n ( ( P e. D /\ F Fn P /\ ph' ) /\ ps' )
16	5, 15	nfxfr 1704	. . . 4 |- F/ n ( P e. D /\ F Fn P /\ ph' /\ ps' )
17		eleq1 2537	. . . . . 6 |- ( n = P -> ( n e. D <-> P e. D ) )
18		fneq2 5675	. . . . . . 7 |- ( n = P -> ( F Fn n <-> F Fn P ) )
19		sbceq1a 3266	. . . . . . . 8 |- ( n = P -> ( ph <-> [. P / n ]. ph ) )
20	19, 8	syl6bbr 271	. . . . . . 7 |- ( n = P -> ( ph <-> ph' ) )
21		sbceq1a 3266	. . . . . . . 8 |- ( n = P -> ( ps <-> [. P / n ]. ps ) )
22	21, 12	syl6bbr 271	. . . . . . 7 |- ( n = P -> ( ps <-> ps' ) )
23	18, 20, 22	3anbi123d 1365	. . . . . 6 |- ( n = P -> ( ( F Fn n /\ ph /\ ps ) <-> ( F Fn P /\ ph' /\ ps' ) ) )
24	17, 23	anbi12d 725	. . . . 5 |- ( n = P -> ( ( n e. D /\ ( F Fn n /\ ph /\ ps ) ) <-> ( P e. D /\ ( F Fn P /\ ph' /\ ps' ) ) ) )
25		bnj252 29580	. . . . 5 |- ( ( n e. D /\ F Fn n /\ ph /\ ps ) <-> ( n e. D /\ ( F Fn n /\ ph /\ ps ) ) )
26		bnj252 29580	. . . . 5 |- ( ( P e. D /\ F Fn P /\ ph' /\ ps' ) <-> ( P e. D /\ ( F Fn P /\ ph' /\ ps' ) ) )
27	24, 25, 26	3bitr4g 296	. . . 4 |- ( n = P -> ( ( n e. D /\ F Fn n /\ ph /\ ps ) <-> ( P e. D /\ F Fn P /\ ph' /\ ps' ) ) )
28	16, 27	sbciegf 3287	. . 3 |- ( P e. _V -> ( [. P / n ]. ( n e. D /\ F Fn n /\ ph /\ ps ) <-> ( P e. D /\ F Fn P /\ ph' /\ ps' ) ) )
29	4, 28	ax-mp 5	. 2 |- ( [. P / n ]. ( n e. D /\ F Fn n /\ ph /\ ps ) <-> ( P e. D /\ F Fn P /\ ph' /\ ps' ) )
30	1, 3, 29	3bitri 279	1 |- ( ch' <-> ( P e. D /\ F Fn P /\ ph' /\ ps' ) )

Syntax hints:   <-> wb 189   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   _Vcvv 3031   [.wsbc 3255   Fn wfn 5584   /\ w-bnj17 29563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451

This theorem depends on definitions:  df-bi 190  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-v 3033  df-sbc 3256  df-fn 5592  df-bnj17 29564

This theorem is referenced by:  bnj910  29831  bnj999  29840  bnj907  29848