Theorem ra4sbc 1987

Description: Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. This provides an axiom for a predicate calculus for a restricted domain. This theorem generalizes the unrestricted stdpc4 1181 and a4sbc 1935. See also ra4sbca 1988 and ra4csbela 2032.

Assertion

Ref	Expression
ra4sbc	|- (A e. B -> (A.x e. B ph -> [A / x]ph))

Distinct variable group:   x,B

Proof of Theorem ra4sbc

Step	Hyp	Ref	Expression
1		dfsbcq 1933	. . . . 5 |- (y = A -> ([y / x](x e. B -> ph) <-> [A / x](x e. B -> ph)))
2		sbcimg 1960	. . . . . . . 8 |- (A e. B -> ([A / x](x e. B -> ph) <-> ([A / x]x e. B -> [A / x]ph)))
3		sbcel1gv 1970	. . . . . . . . 9 |- (A e. B -> ([A / x]x e. B <-> A e. B))
4	3	imbi1d 611	. . . . . . . 8 |- (A e. B -> (([A / x]x e. B -> [A / x]ph) <-> (A e. B -> [A / x]ph)))
5	2, 4	bitrd 526	. . . . . . 7 |- (A e. B -> ([A / x](x e. B -> ph) <-> (A e. B -> [A / x]ph)))
6	5	biimpd 153	. . . . . 6 |- (A e. B -> ([A / x](x e. B -> ph) -> (A e. B -> [A / x]ph)))
7	6	pm2.43b 67	. . . . 5 |- ([A / x](x e. B -> ph) -> (A e. B -> [A / x]ph))
8	1, 7	syl6bi 214	. . . 4 |- (y = A -> ([y / x](x e. B -> ph) -> (A e. B -> [A / x]ph)))
9		df-ral 1641	. . . . 5 |- (A.x e. B ph <-> A.x(x e. B -> ph))
10		stdpc4 1181	. . . . 5 |- (A.x(x e. B -> ph) -> [y / x](x e. B -> ph))
11	9, 10	sylbi 199	. . . 4 |- (A.x e. B ph -> [y / x](x e. B -> ph))
12	8, 11	syl5 21	. . 3 |- (y = A -> (A.x e. B ph -> (A e. B -> [A / x]ph)))
13	12	vtocleg 1846	. 2 |- (A e. B -> (A.x e. B ph -> (A e. B -> [A / x]ph)))
14	13	pm2.43a 66	1 |- (A e. B -> (A.x e. B ph -> [A / x]ph))

Syntax hints:   -> wi 3  A.wal 951   = wceq 953   e. wcel 955  [wsbc 1166  A.wral 1637

This theorem is referenced by:  ra4sbca 1988  ra4esbca 1989  ra4csbela 2032  reuuniss2 2881  fzrevralt 6451

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-ral 1641  df-v 1803  df-sbc 1932

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