Theorem csbopr2g 3989

Description: Move class substitution in and out of an operation.

Assertion

Ref	Expression
csbopr2g	|- (A e. D -> [_A / x]_(BFC) = (BF[_A / x]_C))

Distinct variable groups:   x,B   x,F

Proof of Theorem csbopr2g

Step	Hyp	Ref	Expression
1		csbopr12g 3987	. 2 |- (A e. D -> [_A / x]_(BFC) = ([_A / x]_BF[_A / x]_C))
2		ax-17 971	. . . 4 |- (y e. B -> A.x y e. B)
3	2	csbconstgf 2010	. . 3 |- (A e. D -> [_A / x]_B = B)
4	3	opreq1d 3975	. 2 |- (A e. D -> ([_A / x]_BF[_A / x]_C) = (BF[_A / x]_C))
5	1, 4	eqtrd 1507	1 |- (A e. D -> [_A / x]_(BFC) = (BF[_A / x]_C))

Syntax hints:   -> wi 3   = wceq 956   e. wcel 958  [_csb 2001  (class class class)co 3963

This theorem is referenced by:  fsummulc1 7033  efaddlem5 7342  ipval2lem1 8351

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-xp 3184  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fv 3198  df-opr 3965

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