Theorem ra4sbc 2000

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 1187 and a4sbc 1948. See also ra4sbca 2001 and ra4csbela 2045.

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 1946	. . . . 5 |- (y = A -> ([y / x](x e. B -> ph) <-> [A / x](x e. B -> ph)))
2		sbcimg 1973	. . . . . . . 8 |- (A e. B -> ([A / x](x e. B -> ph) <-> ([A / x]x e. B -> [A / x]ph)))
3		sbcel1gv 1983	. . . . . . . . 9 |- (A e. B -> ([A / x]x e. B <-> A e. B))
4	3	imbi1d 615	. . . . . . . 8 |- (A e. B -> (([A / x]x e. B -> [A / x]ph) <-> (A e. B -> [A / x]ph)))
5	2, 4	bitrd 530	. . . . . . 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 1652	. . . . 5 |- (A.x e. B ph <-> A.x(x e. B -> ph))
10		stdpc4 1187	. . . . 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 1858	. 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 956   = wceq 958   e. wcel 960  [wsbc 1172  A.wral 1648

This theorem is referenced by:  ra4sbca 2001  ra4esbca 2002  ra4csbela 2045  reuuniss2 2897  fzrevralt 6520

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-v 1815  df-sbc 1945

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