Theorem sbciegft 3077

Description: Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 3078.) (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)

Assertion

Ref	Expression
sbciegft	|- ((A e. V /\ F/xps /\ 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 3071	. . 3 |- ( [.A / x ].ph <-> E.x(x = A /\ ph))
2		bi1 178	. . . . . . . 8 |- ((ph <-> ps) -> (ph -> ps))
3	2	imim2i 13	. . . . . . 7 |- ((x = A -> (ph <-> ps)) -> (x = A -> (ph -> ps)))
4	3	imp3a 420	. . . . . 6 |- ((x = A -> (ph <-> ps)) -> ((x = A /\ ph) -> ps))
5	4	alimi 1559	. . . . 5 |- (A.x(x = A -> (ph <-> ps)) -> A.x((x = A /\ ph) -> ps))
6		19.23t 1800	. . . . . 6 |- ( F/xps -> (A.x((x = A /\ ph) -> ps) <-> (E.x(x = A /\ ph) -> ps)))
7	6	biimpa 470	. . . . 5 |- (( F/xps /\ A.x((x = A /\ ph) -> ps)) -> (E.x(x = A /\ ph) -> ps))
8	5, 7	sylan2 460	. . . 4 |- (( F/xps /\ A.x(x = A -> (ph <-> ps))) -> (E.x(x = A /\ ph) -> ps))
9	8	3adant1 973	. . 3 |- ((A e. V /\ F/xps /\ A.x(x = A -> (ph <-> ps))) -> (E.x(x = A /\ ph) -> ps))
10	1, 9	syl5bi 208	. 2 |- ((A e. V /\ F/xps /\ A.x(x = A -> (ph <-> ps))) -> ( [.A / x ].ph -> ps))
11		bi2 189	. . . . . . . 8 |- ((ph <-> ps) -> (ps -> ph))
12	11	imim2i 13	. . . . . . 7 |- ((x = A -> (ph <-> ps)) -> (x = A -> (ps -> ph)))
13	12	com23 72	. . . . . 6 |- ((x = A -> (ph <-> ps)) -> (ps -> (x = A -> ph)))
14	13	alimi 1559	. . . . 5 |- (A.x(x = A -> (ph <-> ps)) -> A.x(ps -> (x = A -> ph)))
15		19.21t 1795	. . . . . 6 |- ( F/xps -> (A.x(ps -> (x = A -> ph)) <-> (ps -> A.x(x = A -> ph))))
16	15	biimpa 470	. . . . 5 |- (( F/xps /\ A.x(ps -> (x = A -> ph))) -> (ps -> A.x(x = A -> ph)))
17	14, 16	sylan2 460	. . . 4 |- (( F/xps /\ A.x(x = A -> (ph <-> ps))) -> (ps -> A.x(x = A -> ph)))
18	17	3adant1 973	. . 3 |- ((A e. V /\ F/xps /\ A.x(x = A -> (ph <-> ps))) -> (ps -> A.x(x = A -> ph)))
19		sbc6g 3072	. . . 4 |- (A e. V -> ( [.A / x ].ph <-> A.x(x = A -> ph)))
20	19	3ad2ant1 976	. . 3 |- ((A e. V /\ F/xps /\ A.x(x = A -> (ph <-> ps))) -> ( [.A / x ].ph <-> A.x(x = A -> ph)))
21	18, 20	sylibrd 225	. 2 |- ((A e. V /\ F/xps /\ A.x(x = A -> (ph <-> ps))) -> (ps -> [.A / x ].ph))
22	10, 21	impbid 183	1 |- ((A e. V /\ F/xps /\ A.x(x = A -> (ph <-> ps))) -> ( [.A / x ].ph <-> ps))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934  A.wal 1540  E.wex 1541   F/wnf 1544   = wceq 1642   e. wcel 1710   [.wsbc 3047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sbc 3048

This theorem is referenced by:  sbciegf  3078  sbciedf  3082