Theorem elrab3t 2935

Description: Membership in a restricted class abstraction, using implicit substitution. (Closed theorem version of elrab3 2937.) (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 1565	. . . . 5 |- F/ x A. x ( x = A -> ( ph <-> ps ) )
3		nfv 1552	. . . . 5 |- F/ x A e. B
4	2, 3	nfan 1589	. . . 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 1582	. . . . 5 |- ( ( A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B ) -> ( x = A -> ( ph <-> ps ) ) )
7		eleq1 2270	. . . . . . . . . 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 1546	. . 3 |- ( ( A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B ) -> A. x ( x = A -> ( ( x e. B /\ ph ) <-> ps ) ) )
15		elabgt 2921	. . 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 2495	. . . 4 |- { x e. B | ph } = { x | ( x e. B /\ ph ) }
18	17	eleq2i 2274	. . 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 1371   = wceq 1373   e. wcel 2178   {cab 2193   {crab 2490

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rab 2495  df-v 2778

This theorem is referenced by:  f1oresrab  5768