Theorem sbc6g 1955

Description: An equivalence for class substitution.

Assertion

Ref	Expression
sbc6g	|- (A e. B -> ([A / x]ph <-> A.x(x = A -> ph)))

Distinct variable group:   x,A

Proof of Theorem sbc6g

Step	Hyp	Ref	Expression
1		biimt 731	. . . . . 6 |- (A e. V -> (ph <-> (A e. V -> ph)))
2	1	imbi2d 612	. . . . 5 |- (A e. V -> ((x = A -> ph) <-> (x = A -> (A e. V -> ph))))
3	2	albidv 1278	. . . 4 |- (A e. V -> (A.x(x = A -> ph) <-> A.x(x = A -> (A e. V -> ph))))
4		biimt 731	. . . 4 |- (A e. V -> (A.x(x = A -> ph) <-> (A e. V -> A.x(x = A -> ph))))
5		ax-17 971	. . . . . 6 |- (y e. A -> A.x y e. A)
6	5	hbsbc1 1949	. . . . 5 |- ((A e. V -> [A / x]ph) -> A.x(A e. V -> [A / x]ph))
7		sbceq1a 1944	. . . . . 6 |- (x = A -> (ph <-> [A / x]ph))
8	7	imbi2d 612	. . . . 5 |- (x = A -> ((A e. V -> ph) <-> (A e. V -> [A / x]ph)))
9	6, 8	ceqsalg 1825	. . . 4 |- (A e. V -> (A.x(x = A -> (A e. V -> ph)) <-> (A e. V -> [A / x]ph)))
10	3, 4, 9	3bitr3rd 549	. . 3 |- (A e. V -> ((A e. V -> [A / x]ph) <-> (A e. V -> A.x(x = A -> ph))))
11	10	pm5.74rd 588	. 2 |- (A e. V -> (A e. V -> ([A / x]ph <-> A.x(x = A -> ph))))
12		elisset 1817	. 2 |- (A e. B -> A e. V)
13	11, 12, 12	sylc 68	1 |- (A e. B -> ([A / x]ph <-> A.x(x = A -> ph)))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 954   = wceq 956   e. wcel 958  [wsbc 1170  Vcvv 1811

This theorem is referenced by:  sbc6 1957  sbciegft 1959  sbcralt 1990  sbcralgf 1992  sbcsng 2753  fz1sbct 6517

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-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-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-sbc 1942

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