Theorem csbopr2g 3995

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 3993	. 2 |- (A e. D -> [_A / x]_(BFC) = ([_A / x]_BF[_A / x]_C))
2		ax-17 973	. . . 4 |- (y e. B -> A.x y e. B)
3	2	csbconstgf 2013	. . 3 |- (A e. D -> [_A / x]_B = B)
4	3	opreq1d 3981	. 2 |- (A e. D -> ([_A / x]_BF[_A / x]_C) = (BF[_A / x]_C))
5	1, 4	eqtrd 1510	1 |- (A e. D -> [_A / x]_(BFC) = (BF[_A / x]_C))

Syntax hints:   -> wi 3   = wceq 958   e. wcel 960  [_csb 2004  (class class class)co 3969

This theorem is referenced by:  fsummulc1 7033  efaddlem5 7342  ipval2lem1 8347

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-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  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  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-xp 3190  df-cnv 3192  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fv 3204  df-opr 3971

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