Theorem sbciegft 1959

Description: Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 1960.)

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 1954	. . . 4 |- (A e. B -> ([A / x]ph <-> E.x(x = A /\ ph)))
2	1	3ad2ant1 800	. . 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 1116	. . . . . 6 |- (A.x(ps -> A.xps) -> (A.x((x = A /\ ph) -> ps) <-> (E.x(x = A /\ ph) -> ps)))
4	3	biimpa 416	. . . . 5 |- ((A.x(ps -> A.xps) /\ A.x((x = A /\ ph) -> ps)) -> (E.x(x = A /\ ph) -> ps))
5		bi1 148	. . . . . . . 8 |- ((ph <-> ps) -> (ph -> ps))
6	5	imim2i 17	. . . . . . 7 |- ((x = A -> (ph <-> ps)) -> (x = A -> (ph -> ps)))
7	6	imp3a 361	. . . . . 6 |- ((x = A -> (ph <-> ps)) -> ((x = A /\ ph) -> ps))
8	7	19.20i 992	. . . . 5 |- (A.x(x = A -> (ph <-> ps)) -> A.x((x = A /\ ph) -> ps))
9	4, 8	sylan2 451	. . . 4 |- ((A.x(ps -> A.xps) /\ A.x(x = A -> (ph <-> ps))) -> (E.x(x = A /\ ph) -> ps))
10	9	3adant1 797	. . 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 203	. 2 |- ((A e. B /\ A.x(ps -> A.xps) /\ A.x(x = A -> (ph <-> ps))) -> ([A / x]ph -> ps))
12		19.21t 1115	. . . . . 6 |- (A.x(ps -> A.xps) -> (A.x(ps -> (x = A -> ph)) <-> (ps -> A.x(x = A -> ph))))
13	12	biimpa 416	. . . . 5 |- ((A.x(ps -> A.xps) /\ A.x(ps -> (x = A -> ph))) -> (ps -> A.x(x = A -> ph)))
14		bi2 149	. . . . . . . 8 |- ((ph <-> ps) -> (ps -> ph))
15	14	imim2i 17	. . . . . . 7 |- ((x = A -> (ph <-> ps)) -> (x = A -> (ps -> ph)))
16	15	com23 32	. . . . . 6 |- ((x = A -> (ph <-> ps)) -> (ps -> (x = A -> ph)))
17	16	19.20i 992	. . . . 5 |- (A.x(x = A -> (ph <-> ps)) -> A.x(ps -> (x = A -> ph)))
18	13, 17	sylan2 451	. . . 4 |- ((A.x(ps -> A.xps) /\ A.x(x = A -> (ph <-> ps))) -> (ps -> A.x(x = A -> ph)))
19	18	3adant1 797	. . 3 |- ((A e. B /\ A.x(ps -> A.xps) /\ A.x(x = A -> (ph <-> ps))) -> (ps -> A.x(x = A -> ph)))
20		sbc6g 1955	. . . 4 |- (A e. B -> ([A / x]ph <-> A.x(x = A -> ph)))
21	20	3ad2ant1 800	. . 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 204	. 2 |- ((A e. B /\ A.x(ps -> A.xps) /\ A.x(x = A -> (ph <-> ps))) -> (ps -> [A / x]ph))
23	11, 22	impbid 516	1 |- ((A e. B /\ A.x(ps -> A.xps) /\ A.x(x = A -> (ph <-> ps))) -> ([A / x]ph <-> ps))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 775  A.wal 954   = wceq 956   e. wcel 958  E.wex 980  [wsbc 1170

This theorem is referenced by:  sbciegf 1960  csbiegft 2029

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-sbc 1942

Copyright terms: Public domain