Theorem csbie2 2037

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

Hypotheses

Ref	Expression
csbie2g.1	|- A e. V
csbie2g.2	|- B e. V
csbie2.3	|- ((x = A /\ y = B) -> C = D)

Assertion

Ref	Expression
csbie2	|- [_A / x]_[_B / y]_C = D

Distinct variable groups:   x,y,A   x,B,y   x,D,y

Proof of Theorem csbie2

Step	Hyp	Ref	Expression
1		csbie2.3	. . 3 |- ((x = A /\ y = B) -> C = D)
2	1	gen2 985	. 2 |- A.xA.y((x = A /\ y = B) -> C = D)
3		csbie2g.1	. . 3 |- A e. V
4		csbie2g.2	. . 3 |- B e. V
5	3, 4	csbie2t 2036	. 2 |- (A.xA.y((x = A /\ y = B) -> C = D) -> [_A / x]_[_B / y]_C = D)
6	2, 5	ax-mp 7	1 |- [_A / x]_[_B / y]_C = D

Syntax hints:   -> wi 3   /\ wa 223  A.wal 956   = wceq 958   e. wcel 960  Vcvv 1814  [_csb 2004

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-sbc 1945  df-csb 2005

Copyright terms: Public domain