Theorem csbco3g 2030

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 2026	. 2 |- ((A e. R /\ A.x B e. S) -> [_A / x]_[_B / y]_C = [_[_A / x]_B / y]_C)
2		ax-17 968	. . . . . 6 |- (z e. D -> A.x z e. D)
3	2	gen2 980	. . . . 5 |- A.xA.z(z e. D -> A.x z e. D)
4		csbco3g.1	. . . . . 6 |- (x = A -> B = D)
5	4	ax-gen 960	. . . . 5 |- A.x(x = A -> B = D)
6		csbiegft 2019	. . . . 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 905	. . . 4 |- (A e. R -> [_A / x]_B = D)
8	7	csbeq1d 1994	. . 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 1499	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 951   = wceq 953   e. wcel 955  [_csb 1991

This theorem is referenced by:  fsumrev 6967  fsumshft 6969  fsum0diag2 7194

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-sbc 1932  df-csb 1992

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