Theorem csbiefOLD 2577

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
csbiefOLD	|- [_A / x]_B = C

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

Proof of Theorem csbiefOLD

Step	Hyp	Ref	Expression
1		csbief.3	. . 3 |- (x = A -> B = C)
2	1	ax-gen 1305	. 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 2574	. 2 |- (A.x(x = A -> B = C) <-> [_A / x]_B = C)
6	2, 5	mpbi 206	1 |- [_A / x]_B = C

Syntax hints:   -> wi 3  A.wal 1296   = wceq 1298   e. wcel 1300  _Vcvv 2292  [_csb 2540

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-sbc 2454  df-csb 2541

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