Theorem isfildlem 20872

Description: Lemma for isfild 20873. (Contributed by Mario Carneiro, 1-Dec-2013.)

Hypotheses

Ref	Expression
isfild.1	|- ( ph -> ( x e. F <-> ( x C_ A /\ ps ) ) )
isfild.2	|- ( ph -> A e. _V )

Assertion

Ref	Expression
isfildlem	|- ( ph -> ( B e. F <-> ( B C_ A /\ [. B / x ]. ps ) ) )

Distinct variable groups:   x, A   x, F   ph, x

Allowed substitution hints:   ps( x)   B( x)

Proof of Theorem isfildlem

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3054	. . 3 |- ( B e. F -> B e. _V )
2	1	a1i 11	. 2 |- ( ph -> ( B e. F -> B e. _V ) )
3		isfild.2	. . . 4 |- ( ph -> A e. _V )
4		ssexg 4549	. . . . 5 |- ( ( B C_ A /\ A e. _V ) -> B e. _V )
5	4	expcom 437	. . . 4 |- ( A e. _V -> ( B C_ A -> B e. _V ) )
6	3, 5	syl 17	. . 3 |- ( ph -> ( B C_ A -> B e. _V ) )
7	6	adantrd 470	. 2 |- ( ph -> ( ( B C_ A /\ [. B / x ]. ps ) -> B e. _V ) )
8		eleq1 2517	. . . . . 6 |- ( y = B -> ( y e. F <-> B e. F ) )
9		sseq1 3453	. . . . . . 7 |- ( y = B -> ( y C_ A <-> B C_ A ) )
10		dfsbcq 3269	. . . . . . 7 |- ( y = B -> ( [. y / x ]. ps <-> [. B / x ]. ps ) )
11	9, 10	anbi12d 717	. . . . . 6 |- ( y = B -> ( ( y C_ A /\ [. y / x ]. ps ) <-> ( B C_ A /\ [. B / x ]. ps ) ) )
12	8, 11	bibi12d 323	. . . . 5 |- ( y = B -> ( ( y e. F <-> ( y C_ A /\ [. y / x ]. ps ) ) <-> ( B e. F <-> ( B C_ A /\ [. B / x ]. ps ) ) ) )
13	12	imbi2d 318	. . . 4 |- ( y = B -> ( ( ph -> ( y e. F <-> ( y C_ A /\ [. y / x ]. ps ) ) ) <-> ( ph -> ( B e. F <-> ( B C_ A /\ [. B / x ]. ps ) ) ) ) )
14		nfv 1761	. . . . . 6 |- F/ x ph
15		nfv 1761	. . . . . . 7 |- F/ x y e. F
16		nfv 1761	. . . . . . . 8 |- F/ x y C_ A
17		nfsbc1v 3287	. . . . . . . 8 |- F/ x [. y / x ]. ps
18	16, 17	nfan 2011	. . . . . . 7 |- F/ x ( y C_ A /\ [. y / x ]. ps )
19	15, 18	nfbi 2017	. . . . . 6 |- F/ x ( y e. F <-> ( y C_ A /\ [. y / x ]. ps ) )
20	14, 19	nfim 2003	. . . . 5 |- F/ x ( ph -> ( y e. F <-> ( y C_ A /\ [. y / x ]. ps ) ) )
21		eleq1 2517	. . . . . . 7 |- ( x = y -> ( x e. F <-> y e. F ) )
22		sseq1 3453	. . . . . . . 8 |- ( x = y -> ( x C_ A <-> y C_ A ) )
23		sbceq1a 3278	. . . . . . . 8 |- ( x = y -> ( ps <-> [. y / x ]. ps ) )
24	22, 23	anbi12d 717	. . . . . . 7 |- ( x = y -> ( ( x C_ A /\ ps ) <-> ( y C_ A /\ [. y / x ]. ps ) ) )
25	21, 24	bibi12d 323	. . . . . 6 |- ( x = y -> ( ( x e. F <-> ( x C_ A /\ ps ) ) <-> ( y e. F <-> ( y C_ A /\ [. y / x ]. ps ) ) ) )
26	25	imbi2d 318	. . . . 5 |- ( x = y -> ( ( ph -> ( x e. F <-> ( x C_ A /\ ps ) ) ) <-> ( ph -> ( y e. F <-> ( y C_ A /\ [. y / x ]. ps ) ) ) ) )
27		isfild.1	. . . . 5 |- ( ph -> ( x e. F <-> ( x C_ A /\ ps ) ) )
28	20, 26, 27	chvar 2106	. . . 4 |- ( ph -> ( y e. F <-> ( y C_ A /\ [. y / x ]. ps ) ) )
29	13, 28	vtoclg 3107	. . 3 |- ( B e. _V -> ( ph -> ( B e. F <-> ( B C_ A /\ [. B / x ]. ps ) ) ) )
30	29	com12 32	. 2 |- ( ph -> ( B e. _V -> ( B e. F <-> ( B C_ A /\ [. B / x ]. ps ) ) ) )
31	2, 7, 30	pm5.21ndd 356	1 |- ( ph -> ( B e. F <-> ( B C_ A /\ [. B / x ]. ps ) ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1444   e. wcel 1887   _Vcvv 3045   [.wsbc 3267   C_ wss 3404

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525

This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-v 3047  df-sbc 3268  df-in 3411  df-ss 3418

This theorem is referenced by:  isfild  20873