Theorem csbief 2028

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 961	. 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 2026	. 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 952   = wceq 954   e. wcel 956  Vcvv 1807  [_csb 1997

This theorem is referenced by:  eqerlem 4260  binomlem1 7012  binomlem2 7013  binomlem4 7015  iserzshft2 7052

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-sbc 1938  df-csb 1998

Copyright terms: Public domain