Theorem elrab3t 2919

Description: Membership in a restricted class abstraction, using implicit substitution. (Closed theorem version of elrab3 2921.) (Contributed by Thierry Arnoux, 31-Aug-2017.)

Assertion

Ref	Expression
elrab3t	|- ( ( A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B ) -> ( A e. { x e. B | ph } <-> ps ) )

Distinct variable groups:   x, A   x, B   ps, x

Allowed substitution hint:   ph( x)

Proof of Theorem elrab3t

Step	Hyp	Ref	Expression
1		simpr 110	. . 3 |- ( ( A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B ) -> A e. B )
2		nfa1 1555	. . . . 5 |- F/ x A. x ( x = A -> ( ph <-> ps ) )
3		nfv 1542	. . . . 5 |- F/ x A e. B
4	2, 3	nfan 1579	. . . 4 |- F/ x ( A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B )
5		simpl 109	. . . . . 6 |- ( ( A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B ) -> A. x ( x = A -> ( ph <-> ps ) ) )
6	5	19.21bi 1572	. . . . 5 |- ( ( A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B ) -> ( x = A -> ( ph <-> ps ) ) )
7		eleq1 2259	. . . . . . . . . 10 |- ( x = A -> ( x e. B <-> A e. B ) )
8	7	biimparc 299	. . . . . . . . 9 |- ( ( A e. B /\ x = A ) -> x e. B )
9	8	biantrurd 305	. . . . . . . 8 |- ( ( A e. B /\ x = A ) -> ( ph <-> ( x e. B /\ ph ) ) )
10	9	bibi1d 233	. . . . . . 7 |- ( ( A e. B /\ x = A ) -> ( ( ph <-> ps ) <-> ( ( x e. B /\ ph ) <-> ps ) ) )
11	10	pm5.74da 443	. . . . . 6 |- ( A e. B -> ( ( x = A -> ( ph <-> ps ) ) <-> ( x = A -> ( ( x e. B /\ ph ) <-> ps ) ) ) )
12	11	adantl 277	. . . . 5 |- ( ( A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B ) -> ( ( x = A -> ( ph <-> ps ) ) <-> ( x = A -> ( ( x e. B /\ ph ) <-> ps ) ) ) )
13	6, 12	mpbid 147	. . . 4 |- ( ( A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B ) -> ( x = A -> ( ( x e. B /\ ph ) <-> ps ) ) )
14	4, 13	alrimi 1536	. . 3 |- ( ( A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B ) -> A. x ( x = A -> ( ( x e. B /\ ph ) <-> ps ) ) )
15		elabgt 2905	. . 3 |- ( ( A e. B /\ A. x ( x = A -> ( ( x e. B /\ ph ) <-> ps ) ) ) -> ( A e. { x | ( x e. B /\ ph ) } <-> ps ) )
16	1, 14, 15	syl2anc 411	. 2 |- ( ( A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B ) -> ( A e. { x | ( x e. B /\ ph ) } <-> ps ) )
17		df-rab 2484	. . . 4 |- { x e. B | ph } = { x | ( x e. B /\ ph ) }
18	17	eleq2i 2263	. . 3 |- ( A e. { x e. B | ph } <-> A e. { x | ( x e. B /\ ph ) } )
19	18	bibi1i 228	. 2 |- ( ( A e. { x e. B | ph } <-> ps ) <-> ( A e. { x | ( x e. B /\ ph ) } <-> ps ) )
20	16, 19	sylibr 134	1 |- ( ( A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B ) -> ( A e. { x e. B | ph } <-> ps ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1362   = wceq 1364   e. wcel 2167   {cab 2182   {crab 2479

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rab 2484  df-v 2765

This theorem is referenced by:  f1oresrab  5730