Theorem sbciegft 2981

Description: Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 2982.) (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)

Assertion

Ref	Expression
sbciegft	|- ( ( A e. V /\ F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) ) -> ( [. A / x ]. ph <-> ps ) )

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   ps( x)   V( x)

Proof of Theorem sbciegft

Step	Hyp	Ref	Expression
1		sbc5 2974	. . 3 |- ( [. A / x ]. ph <-> E. x ( x = A /\ ph ) )
2		biimp 117	. . . . . . . 8 |- ( ( ph <-> ps ) -> ( ph -> ps ) )
3	2	imim2i 12	. . . . . . 7 |- ( ( x = A -> ( ph <-> ps ) ) -> ( x = A -> ( ph -> ps ) ) )
4	3	impd 252	. . . . . 6 |- ( ( x = A -> ( ph <-> ps ) ) -> ( ( x = A /\ ph ) -> ps ) )
5	4	alimi 1443	. . . . 5 |- ( A. x ( x = A -> ( ph <-> ps ) ) -> A. x ( ( x = A /\ ph ) -> ps ) )
6		19.23t 1665	. . . . . 6 |- ( F/ x ps -> ( A. x ( ( x = A /\ ph ) -> ps ) <-> ( E. x ( x = A /\ ph ) -> ps ) ) )
7	6	biimpa 294	. . . . 5 |- ( ( F/ x ps /\ A. x ( ( x = A /\ ph ) -> ps ) ) -> ( E. x ( x = A /\ ph ) -> ps ) )
8	5, 7	sylan2 284	. . . 4 |- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) ) -> ( E. x ( x = A /\ ph ) -> ps ) )
9	8	3adant1 1005	. . 3 |- ( ( A e. V /\ F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) ) -> ( E. x ( x = A /\ ph ) -> ps ) )
10	1, 9	syl5bi 151	. 2 |- ( ( A e. V /\ F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) ) -> ( [. A / x ]. ph -> ps ) )
11		biimpr 129	. . . . . . . 8 |- ( ( ph <-> ps ) -> ( ps -> ph ) )
12	11	imim2i 12	. . . . . . 7 |- ( ( x = A -> ( ph <-> ps ) ) -> ( x = A -> ( ps -> ph ) ) )
13	12	com23 78	. . . . . 6 |- ( ( x = A -> ( ph <-> ps ) ) -> ( ps -> ( x = A -> ph ) ) )
14	13	alimi 1443	. . . . 5 |- ( A. x ( x = A -> ( ph <-> ps ) ) -> A. x ( ps -> ( x = A -> ph ) ) )
15		19.21t 1570	. . . . . 6 |- ( F/ x ps -> ( A. x ( ps -> ( x = A -> ph ) ) <-> ( ps -> A. x ( x = A -> ph ) ) ) )
16	15	biimpa 294	. . . . 5 |- ( ( F/ x ps /\ A. x ( ps -> ( x = A -> ph ) ) ) -> ( ps -> A. x ( x = A -> ph ) ) )
17	14, 16	sylan2 284	. . . 4 |- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) ) -> ( ps -> A. x ( x = A -> ph ) ) )
18	17	3adant1 1005	. . 3 |- ( ( A e. V /\ F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) ) -> ( ps -> A. x ( x = A -> ph ) ) )
19		sbc6g 2975	. . . 4 |- ( A e. V -> ( [. A / x ]. ph <-> A. x ( x = A -> ph ) ) )
20	19	3ad2ant1 1008	. . 3 |- ( ( A e. V /\ F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) ) -> ( [. A / x ]. ph <-> A. x ( x = A -> ph ) ) )
21	18, 20	sylibrd 168	. 2 |- ( ( A e. V /\ F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) ) -> ( ps -> [. A / x ]. ph ) )
22	10, 21	impbid 128	1 |- ( ( A e. V /\ F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) ) -> ( [. A / x ]. ph <-> ps ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 968   A.wal 1341   = wceq 1343   F/wnf 1448   E.wex 1480   e. wcel 2136   [.wsbc 2951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-sbc 2952

This theorem is referenced by:  sbciegf  2982  sbciedf  2986