Theorem bnj919 29366

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 3361	. 2 |- ( [. P / n ]. ch <-> [. P / n ]. ( n e. D /\ F Fn n /\ ph /\ ps ) )
4		bnj919.5	. . 3 |- P e. _V
5		df-bnj17 29280	. . . . 5 |- ( ( P e. D /\ F Fn P /\ ph' /\ ps' ) <-> ( ( P e. D /\ F Fn P /\ ph' ) /\ ps' ) )
6		nfv 1754	. . . . . . 7 |- F/ n P e. D
7		nfv 1754	. . . . . . 7 |- F/ n F Fn P
8		bnj919.2	. . . . . . . 8 |- ( ph' <-> [. P / n ]. ph )
9		nfsbc1v 3325	. . . . . . . 8 |- F/ n [. P / n ]. ph
10	8, 9	nfxfr 1692	. . . . . . 7 |- F/ n ph'
11	6, 7, 10	nf3an 1988	. . . . . 6 |- F/ n ( P e. D /\ F Fn P /\ ph' )
12		bnj919.3	. . . . . . 7 |- ( ps' <-> [. P / n ]. ps )
13		nfsbc1v 3325	. . . . . . 7 |- F/ n [. P / n ]. ps
14	12, 13	nfxfr 1692	. . . . . 6 |- F/ n ps'
15	11, 14	nfan 1986	. . . . 5 |- F/ n ( ( P e. D /\ F Fn P /\ ph' ) /\ ps' )
16	5, 15	nfxfr 1692	. . . 4 |- F/ n ( P e. D /\ F Fn P /\ ph' /\ ps' )
17		eleq1 2501	. . . . . 6 |- ( n = P -> ( n e. D <-> P e. D ) )
18		fneq2 5683	. . . . . . 7 |- ( n = P -> ( F Fn n <-> F Fn P ) )
19		sbceq1a 3316	. . . . . . . 8 |- ( n = P -> ( ph <-> [. P / n ]. ph ) )
20	19, 8	syl6bbr 266	. . . . . . 7 |- ( n = P -> ( ph <-> ph' ) )
21		sbceq1a 3316	. . . . . . . 8 |- ( n = P -> ( ps <-> [. P / n ]. ps ) )
22	21, 12	syl6bbr 266	. . . . . . 7 |- ( n = P -> ( ps <-> ps' ) )
23	18, 20, 22	3anbi123d 1335	. . . . . 6 |- ( n = P -> ( ( F Fn n /\ ph /\ ps ) <-> ( F Fn P /\ ph' /\ ps' ) ) )
24	17, 23	anbi12d 715	. . . . 5 |- ( n = P -> ( ( n e. D /\ ( F Fn n /\ ph /\ ps ) ) <-> ( P e. D /\ ( F Fn P /\ ph' /\ ps' ) ) ) )
25		bnj252 29296	. . . . 5 |- ( ( n e. D /\ F Fn n /\ ph /\ ps ) <-> ( n e. D /\ ( F Fn n /\ ph /\ ps ) ) )
26		bnj252 29296	. . . . 5 |- ( ( P e. D /\ F Fn P /\ ph' /\ ps' ) <-> ( P e. D /\ ( F Fn P /\ ph' /\ ps' ) ) )
27	24, 25, 26	3bitr4g 291	. . . 4 |- ( n = P -> ( ( n e. D /\ F Fn n /\ ph /\ ps ) <-> ( P e. D /\ F Fn P /\ ph' /\ ps' ) ) )
28	16, 27	sbciegf 3337	. . 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 274	1 |- ( ch' <-> ( P e. D /\ F Fn P /\ ph' /\ ps' ) )

Syntax hints:   <-> wb 187   /\ wa 370   /\ w3a 982   = wceq 1437   e. wcel 1870   _Vcvv 3087   [.wsbc 3305   Fn wfn 5596   /\ w-bnj17 29279

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 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407

This theorem depends on definitions:  df-bi 188  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-v 3089  df-sbc 3306  df-fn 5604  df-bnj17 29280

This theorem is referenced by:  bnj910  29547  bnj999  29556  bnj907  29564