Theorem bnj55 12430

Description: First-order logic and set theory.

Hypotheses

Ref	Expression
bnj55.1	|- (ch <-> A.x e. A (ps -> ph))
bnj55.2	|- B C_ A

Assertion

Ref	Expression
bnj55	|- ((ch /\ E.y e. B [y / x]ps) -> E.y e. B [y / x]ph)

Distinct variable groups:   x,A,y   ph,y   ps,y

Proof of Theorem bnj55

Step	Hyp	Ref	Expression
1		df-ral 2109	. . . . . 6 |- (A.x e. A (ps -> ph) <-> A.x(x e. A -> (ps -> ph)))
2		ax-17 1317	. . . . . . 7 |- ((x e. A -> (ps -> ph)) -> A.y(x e. A -> (ps -> ph)))
3	2	sb8 1637	. . . . . 6 |- (A.x(x e. A -> (ps -> ph)) <-> A.y[y / x](x e. A -> (ps -> ph)))
4		bnj45 12415	. . . . . . . 8 |- ([y / x](x e. A -> (ps -> ph)) <-> (y e. A -> [y / x](ps -> ph)))
5		sbim 1604	. . . . . . . . 9 |- ([y / x](ps -> ph) <-> ([y / x]ps -> [y / x]ph))
6	5	imbi2i 202	. . . . . . . 8 |- ((y e. A -> [y / x](ps -> ph)) <-> (y e. A -> ([y / x]ps -> [y / x]ph)))
7	4, 6	bitri 190	. . . . . . 7 |- ([y / x](x e. A -> (ps -> ph)) <-> (y e. A -> ([y / x]ps -> [y / x]ph)))
8	7	albii 1346	. . . . . 6 |- (A.y[y / x](x e. A -> (ps -> ph)) <-> A.y(y e. A -> ([y / x]ps -> [y / x]ph)))
9	1, 3, 8	3bitri 194	. . . . 5 |- (A.x e. A (ps -> ph) <-> A.y(y e. A -> ([y / x]ps -> [y / x]ph)))
10		bnj55.1	. . . . 5 |- (ch <-> A.x e. A (ps -> ph))
11		df-ral 2109	. . . . 5 |- (A.y e. A ([y / x]ps -> [y / x]ph) <-> A.y(y e. A -> ([y / x]ps -> [y / x]ph)))
12	9, 10, 11	3bitr4i 200	. . . 4 |- (ch <-> A.y e. A ([y / x]ps -> [y / x]ph))
13		bnj55.2	. . . . 5 |- B C_ A
14	13	bnj49 12421	. . . 4 |- (A.y e. A ([y / x]ps -> [y / x]ph) -> A.y e. B ([y / x]ps -> [y / x]ph))
15	12, 14	sylbi 216	. . 3 |- (ch -> A.y e. B ([y / x]ps -> [y / x]ph))
16		rexim 2194	. . 3 |- (A.y e. B ([y / x]ps -> [y / x]ph) -> (E.y e. B [y / x]ps -> E.y e. B [y / x]ph))
17	15, 16	syl 12	. 2 |- (ch -> (E.y e. B [y / x]ps -> E.y e. B [y / x]ph))
18	17	imp 377	1 |- ((ch /\ E.y e. B [y / x]ps) -> E.y e. B [y / x]ph)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   e. wcel 1300  [wsbc 1534  A.wral 2105  E.wrex 2106   C_ wss 2593

This theorem is referenced by:  bnj56 13195

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-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-rex 2110  df-in 2603  df-ss 2605

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