Theorem csbief 2003

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 955	. 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 2001	. 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 950   = wceq 1099   e. wcel 1105  Vcvv 1786  [_csb 1972

This theorem is referenced by:  eqerlem 4208  binomlem1 6955  binomlem2 6956  binomlem4 6958  iserzshft2 6995

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 957  df-sb 1155  df-clab 1441  df-cleq 1446  df-clel 1449  df-v 1787  df-sbc 1913  df-csb 1973

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