Theorem sbciegft 3033

Description: Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 3034.) (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 3026	. . 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 1479	. . . . 5 |- ( A. x ( x = A -> ( ph <-> ps ) ) -> A. x ( ( x = A /\ ph ) -> ps ) )
6		19.23t 1701	. . . . . 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 1018	. . 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 1479	. . . . 5 |- ( A. x ( x = A -> ( ph <-> ps ) ) -> A. x ( ps -> ( x = A -> ph ) ) )
15		19.21t 1606	. . . . . 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 1018	. . 3 |- ( ( A e. V /\ F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) ) -> ( ps -> A. x ( x = A -> ph ) ) )
19		sbc6g 3027	. . . 4 |- ( A e. V -> ( [. A / x ]. ph <-> A. x ( x = A -> ph ) ) )
20	19	3ad2ant1 1021	. . 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 981   A.wal 1371   = wceq 1373   F/wnf 1484   E.wex 1516   e. wcel 2177   [.wsbc 3002

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 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-sbc 3003

This theorem is referenced by:  sbciegf  3034  sbciedf  3038