Theorem csbco3g 2011

Description: Composition of two class substitutions.

Hypothesis

Ref	Expression
csbco3g.1	|- (x = A -> B = D)

Assertion

Ref	Expression
csbco3g	|- ((A e. R /\ A.x B e. S) -> [_A / x]_[_B / y]_C = [_D / y]_C)

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

Proof of Theorem csbco3g

Step	Hyp	Ref	Expression
1		csbnestg 2007	. 2 |- ((A e. R /\ A.x B e. S) -> [_A / x]_[_B / y]_C = [_[_A / x]_B / y]_C)
2		ax-17 1190	. . . . . 6 |- (z e. D -> A.x z e. D)
3	2	gen2 959	. . . . 5 |- A.xA.z(z e. D -> A.x z e. D)
4		csbco3g.1	. . . . . 6 |- (x = A -> B = D)
5	4	ax-gen 955	. . . . 5 |- A.x(x = A -> B = D)
6		csbiegft 2000	. . . . 5 |- ((A e. R /\ A.xA.z(z e. D -> A.x z e. D) /\ A.x(x = A -> B = D)) -> [_A / x]_B = D)
7	3, 5, 6	mp3an23 904	. . . 4 |- (A e. R -> [_A / x]_B = D)
8	7	csbeq1d 1975	. . 3 |- (A e. R -> [_[_A / x]_B / y]_C = [_D / y]_C)
9	8	adantr 389	. 2 |- ((A e. R /\ A.x B e. S) -> [_[_A / x]_B / y]_C = [_D / y]_C)
10	1, 9	eqtrd 1483	1 |- ((A e. R /\ A.x B e. S) -> [_A / x]_[_B / y]_C = [_D / y]_C)

Syntax hints:   -> wi 3   /\ wa 223  A.wal 950   = wceq 1099   e. wcel 1105  [_csb 1972

This theorem is referenced by:  fsumrev 6918  fsumshft 6920  fsum0diag2 7145

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-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 774  df-ex 957  df-sb 1155  df-clab 1441  df-cleq 1446  df-clel 1449  df-v 1787  df-sbc 1913  df-csb 1973

Copyright terms: Public domain