Theorem csbief 2032

Description: Conversion of implicit substitution to explicit substitution into a class.

Hypotheses

Ref	Expression
csbief.1	|- A e. V
csbief.2	|- (y e. C -> A.x y e. C)
csbief.3	|- (x = A -> B = C)

Assertion

Ref	Expression
csbief	|- [_A / x]_B = C

Distinct variable groups:   x,A   y,C   x,y

Proof of Theorem csbief

Step	Hyp	Ref	Expression
1		csbief.3	. . 3 |- (x = A -> B = C)
2	1	ax-gen 963	. 2 |- A.x(x = A -> B = C)
3		csbief.1	. . 3 |- A e. V
4		csbief.2	. . 3 |- (y e. C -> A.x y e. C)
5	3, 4	csbieb 2030	. 2 |- (A.x(x = A -> B = C) <-> [_A / x]_B = C)
6	2, 5	mpbi 189	1 |- [_A / x]_B = C

Syntax hints:   -> wi 3  A.wal 954   = wceq 956   e. wcel 958  Vcvv 1811  [_csb 2001

This theorem is referenced by:  eqerlem 4270  binomlem1 7066  binomlem2 7067  binomlem4 7069  iserzshft2 7107

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-8 964  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  df-csb 2002

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