Theorem csbidmg 2029

Description: Idempotent law for class substitutions.

Assertion

Ref	Expression
csbidmg	|- (A e. C -> [_A / x]_[_A / x]_B = [_A / x]_B)

Distinct variable group:   x,A

Proof of Theorem csbidmg

Step	Hyp	Ref	Expression
1		elisset 1808	. 2 |- (A e. C -> A e. V)
2		ax-17 968	. . . 4 |- (A e. V -> A.x A e. V)
3		csbnest1g 2027	. . . 4 |- ((A e. V /\ A.x A e. V) -> [_A / x]_[_A / x]_B = [_[_A / x]_A / x]_B)
4	2, 3	mpdan 702	. . 3 |- (A e. V -> [_A / x]_[_A / x]_B = [_[_A / x]_A / x]_B)
5		ax-17 968	. . . . 5 |- (y e. A -> A.x y e. A)
6	5	csbconstgf 2000	. . . 4 |- (A e. V -> [_A / x]_A = A)
7	6	csbeq1d 1994	. . 3 |- (A e. V -> [_[_A / x]_A / x]_B = [_A / x]_B)
8	4, 7	eqtrd 1499	. 2 |- (A e. V -> [_A / x]_[_A / x]_B = [_A / x]_B)
9	1, 8	syl 10	1 |- (A e. C -> [_A / x]_[_A / x]_B = [_A / x]_B)

Syntax hints:   -> wi 3  A.wal 951   = wceq 953   e. wcel 955  Vcvv 1802  [_csb 1991

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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