Theorem sbciegft 3016

Description: Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 3017.) (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 3009	. . 3 |- ( [. A / x ]. ph <-> E. x ( x = A /\ ph ) )
2		biimp 118	. . . . . . . 8 |- ( ( ph <-> ps ) -> ( ph -> ps ) )
3	2	imim2i 12	. . . . . . 7 |- ( ( x = A -> ( ph <-> ps ) ) -> ( x = A -> ( ph -> ps ) ) )
4	3	impd 254	. . . . . 6 |- ( ( x = A -> ( ph <-> ps ) ) -> ( ( x = A /\ ph ) -> ps ) )
5	4	alimi 1466	. . . . 5 |- ( A. x ( x = A -> ( ph <-> ps ) ) -> A. x ( ( x = A /\ ph ) -> ps ) )
6		19.23t 1688	. . . . . 6 |- ( F/ x ps -> ( A. x ( ( x = A /\ ph ) -> ps ) <-> ( E. x ( x = A /\ ph ) -> ps ) ) )
7	6	biimpa 296	. . . . 5 |- ( ( F/ x ps /\ A. x ( ( x = A /\ ph ) -> ps ) ) -> ( E. x ( x = A /\ ph ) -> ps ) )
8	5, 7	sylan2 286	. . . 4 |- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) ) -> ( E. x ( x = A /\ ph ) -> ps ) )
9	8	3adant1 1017	. . 3 |- ( ( A e. V /\ F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) ) -> ( E. x ( x = A /\ ph ) -> ps ) )
10	1, 9	biimtrid 152	. 2 |- ( ( A e. V /\ F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) ) -> ( [. A / x ]. ph -> ps ) )
11		biimpr 130	. . . . . . . 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 1466	. . . . 5 |- ( A. x ( x = A -> ( ph <-> ps ) ) -> A. x ( ps -> ( x = A -> ph ) ) )
15		19.21t 1593	. . . . . 6 |- ( F/ x ps -> ( A. x ( ps -> ( x = A -> ph ) ) <-> ( ps -> A. x ( x = A -> ph ) ) ) )
16	15	biimpa 296	. . . . 5 |- ( ( F/ x ps /\ A. x ( ps -> ( x = A -> ph ) ) ) -> ( ps -> A. x ( x = A -> ph ) ) )
17	14, 16	sylan2 286	. . . 4 |- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) ) -> ( ps -> A. x ( x = A -> ph ) ) )
18	17	3adant1 1017	. . 3 |- ( ( A e. V /\ F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) ) -> ( ps -> A. x ( x = A -> ph ) ) )
19		sbc6g 3010	. . . 4 |- ( A e. V -> ( [. A / x ]. ph <-> A. x ( x = A -> ph ) ) )
20	19	3ad2ant1 1020	. . 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 169	. 2 |- ( ( A e. V /\ F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) ) -> ( ps -> [. A / x ]. ph ) )
22	10, 21	impbid 129	1 |- ( ( A e. V /\ F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) ) -> ( [. A / x ]. ph <-> ps ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   A.wal 1362   = wceq 1364   F/wnf 1471   E.wex 1503   e. wcel 2164   [.wsbc 2985

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-sbc 2986

This theorem is referenced by:  sbciegf  3017  sbciedf  3021