Theorem sbciegft 2482

Description: Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 2483.)

Assertion

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

Distinct variable group:   x,A

Proof of Theorem sbciegft

Step	Hyp	Ref	Expression
1		sbc5g 2470	. . . 4 |- (A e. B -> ([A / x]ph <-> E.x(x = A /\ ph)))
2	1	3ad2ant1 897	. . 3 |- ((A e. B /\ A.x(ps -> A.xps) /\ A.x(x = A -> (ph <-> ps))) -> ([A / x]ph <-> E.x(x = A /\ ph)))
3		19.23t 1474	. . . . . 6 |- (A.x(ps -> A.xps) -> (A.x((x = A /\ ph) -> ps) <-> (E.x(x = A /\ ph) -> ps)))
4	3	biimpa 460	. . . . 5 |- ((A.x(ps -> A.xps) /\ A.x((x = A /\ ph) -> ps)) -> (E.x(x = A /\ ph) -> ps))
5		bi1 165	. . . . . . . 8 |- ((ph <-> ps) -> (ph -> ps))
6	5	imim2i 11	. . . . . . 7 |- ((x = A -> (ph <-> ps)) -> (x = A -> (ph -> ps)))
7	6	imp3a 388	. . . . . 6 |- ((x = A -> (ph <-> ps)) -> ((x = A /\ ph) -> ps))
8	7	alimi 1338	. . . . 5 |- (A.x(x = A -> (ph <-> ps)) -> A.x((x = A /\ ph) -> ps))
9	4, 8	sylan2 500	. . . 4 |- ((A.x(ps -> A.xps) /\ A.x(x = A -> (ph <-> ps))) -> (E.x(x = A /\ ph) -> ps))
10	9	3adant1 894	. . 3 |- ((A e. B /\ A.x(ps -> A.xps) /\ A.x(x = A -> (ph <-> ps))) -> (E.x(x = A /\ ph) -> ps))
11	2, 10	sylbid 220	. 2 |- ((A e. B /\ A.x(ps -> A.xps) /\ A.x(x = A -> (ph <-> ps))) -> ([A / x]ph -> ps))
12		19.21t 1473	. . . . . 6 |- (A.x(ps -> A.xps) -> (A.x(ps -> (x = A -> ph)) <-> (ps -> A.x(x = A -> ph))))
13	12	biimpa 460	. . . . 5 |- ((A.x(ps -> A.xps) /\ A.x(ps -> (x = A -> ph))) -> (ps -> A.x(x = A -> ph)))
14		bi2 166	. . . . . . . 8 |- ((ph <-> ps) -> (ps -> ph))
15	14	imim2i 11	. . . . . . 7 |- ((x = A -> (ph <-> ps)) -> (x = A -> (ps -> ph)))
16	15	com23 36	. . . . . 6 |- ((x = A -> (ph <-> ps)) -> (ps -> (x = A -> ph)))
17	16	alimi 1338	. . . . 5 |- (A.x(x = A -> (ph <-> ps)) -> A.x(ps -> (x = A -> ph)))
18	13, 17	sylan2 500	. . . 4 |- ((A.x(ps -> A.xps) /\ A.x(x = A -> (ph <-> ps))) -> (ps -> A.x(x = A -> ph)))
19	18	3adant1 894	. . 3 |- ((A e. B /\ A.x(ps -> A.xps) /\ A.x(x = A -> (ph <-> ps))) -> (ps -> A.x(x = A -> ph)))
20		sbc6g 2472	. . . 4 |- (A e. B -> ([A / x]ph <-> A.x(x = A -> ph)))
21	20	3ad2ant1 897	. . 3 |- ((A e. B /\ A.x(ps -> A.xps) /\ A.x(x = A -> (ph <-> ps))) -> ([A / x]ph <-> A.x(x = A -> ph)))
22	19, 21	sylibrd 221	. 2 |- ((A e. B /\ A.x(ps -> A.xps) /\ A.x(x = A -> (ph <-> ps))) -> (ps -> [A / x]ph))
23	11, 22	impbid 574	1 |- ((A e. B /\ A.x(ps -> A.xps) /\ A.x(x = A -> (ph <-> ps))) -> ([A / x]ph <-> ps))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  [wsbc 1534

This theorem is referenced by:  sbciegf 2483  csbiegft 2573  bnj897 12817  bnj920 12827

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-sbc 2454

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