Theorem sbcrext 2361

Description: Interchange class substitution and restricted existential quantifier. (Unnecessary distinct variable restrictions were removed by David Abernethy, 22-Feb-2010.)

Assertion

Ref	Expression
sbcrext	|- (A.y(A e. C /\ A.z(z e. A -> A.y z e. A)) -> ([A / x]E.y e. B ph <-> E.y e. B [A / x]ph))

Distinct variable groups:   z,A   x,B   x,y   y,z

Proof of Theorem sbcrext

Step	Hyp	Ref	Expression
1		dfrex2 1950	. . . . . . 7 |- (E.y e. B ph <-> -. A.y e. B -. ph)
2	1	sbcbii 2339	. . . . . 6 |- (A e. C -> ([A / x]E.y e. B ph <-> [A / x] -. A.y e. B -. ph))
3		sbcng 2328	. . . . . 6 |- (A e. C -> ([A / x] -. A.y e. B -. ph <-> -. [A / x]A.y e. B -. ph))
4	2, 3	bitrd 584	. . . . 5 |- (A e. C -> ([A / x]E.y e. B ph <-> -. [A / x]A.y e. B -. ph))
5	4	adantr 423	. . . 4 |- ((A e. C /\ A.z(z e. A -> A.y z e. A)) -> ([A / x]E.y e. B ph <-> -. [A / x]A.y e. B -. ph))
6	5	a4s 1168	. . 3 |- (A.y(A e. C /\ A.z(z e. A -> A.y z e. A)) -> ([A / x]E.y e. B ph <-> -. [A / x]A.y e. B -. ph))
7		sbcralt 2360	. . . . 5 |- (A.y(A e. C /\ A.z(z e. A -> A.y z e. A)) -> ([A / x]A.y e. B -. ph <-> A.y e. B [A / x] -. ph))
8		hba1 1188	. . . . . 6 |- (A.y(A e. C /\ A.z(z e. A -> A.y z e. A)) -> A.yA.y(A e. C /\ A.z(z e. A -> A.y z e. A)))
9		sbcng 2328	. . . . . . . 8 |- (A e. C -> ([A / x] -. ph <-> -. [A / x]ph))
10	9	adantr 423	. . . . . . 7 |- ((A e. C /\ A.z(z e. A -> A.y z e. A)) -> ([A / x] -. ph <-> -. [A / x]ph))
11	10	a4s 1168	. . . . . 6 |- (A.y(A e. C /\ A.z(z e. A -> A.y z e. A)) -> ([A / x] -. ph <-> -. [A / x]ph))
12	8, 11	ralbid 1955	. . . . 5 |- (A.y(A e. C /\ A.z(z e. A -> A.y z e. A)) -> (A.y e. B [A / x] -. ph <-> A.y e. B -. [A / x]ph))
13	7, 12	bitrd 584	. . . 4 |- (A.y(A e. C /\ A.z(z e. A -> A.y z e. A)) -> ([A / x]A.y e. B -. ph <-> A.y e. B -. [A / x]ph))
14	13	notbid 670	. . 3 |- (A.y(A e. C /\ A.z(z e. A -> A.y z e. A)) -> (-. [A / x]A.y e. B -. ph <-> -. A.y e. B -. [A / x]ph))
15	6, 14	bitrd 584	. 2 |- (A.y(A e. C /\ A.z(z e. A -> A.y z e. A)) -> ([A / x]E.y e. B ph <-> -. A.y e. B -. [A / x]ph))
16		dfrex2 1950	. 2 |- (E.y e. B [A / x]ph <-> -. A.y e. B -. [A / x]ph)
17	15, 16	syl6bbr 594	1 |- (A.y(A e. C /\ A.z(z e. A -> A.y z e. A)) -> ([A / x]E.y e. B ph <-> E.y e. B [A / x]ph))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 162   /\ wa 239  A.wal 1134   e. wcel 1138  [wsbc 1372  A.wral 1939  E.wrex 1940

This theorem is referenced by:  sbcrexgOLD 2367

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1140  ax-7 1142  ax-gen 1143  ax-8 1144  ax-9 1145  ax-10 1146  ax-11 1147  ax-12 1148  ax-17 1155  ax-4 1157  ax-5o 1159  ax-6o 1162  ax-9o 1319  ax-10o 1338  ax-16 1418  ax-11o 1426  ax-ext 1702

This theorem depends on definitions:  df-bi 163  df-or 240  df-an 241  df-ex 1165  df-sb 1374  df-clab 1709  df-cleq 1714  df-clel 1717  df-ral 1943  df-rex 1944  df-v 2127  df-sbc 2287

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