Theorem csbconstgf 2013

Description: Substitution doesn't affect a constant B (in which x is not free).

Hypothesis

Ref	Expression
csbconstgf.1	|- (y e. B -> A.x y e. B)

Assertion

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

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

Proof of Theorem csbconstgf

Step	Hyp	Ref	Expression
1		csbconstgf.1	. . . . 5 |- (y e. B -> A.x y e. B)
2	1	sbcgf 1989	. . . 4 |- (A e. C -> ([A / x]y e. B <-> y e. B))
3	2	abbidv 1580	. . 3 |- (A e. C -> {y | [A / x]y e. B} = {y | y e. B})
4		abid2 1583	. . 3 |- {y | y e. B} = B
5	3, 4	syl6eq 1526	. 2 |- (A e. C -> {y | [A / x]y e. B} = B)
6		df-csb 2005	. 2 |- [_A / x]_B = {y | [A / x]y e. B}
7	5, 6	syl5eq 1522	1 |- (A e. C -> [_A / x]_B = B)

Syntax hints:   -> wi 3  A.wal 956   = wceq 958   e. wcel 960  [wsbc 1172  {cab 1466  [_csb 2004

This theorem is referenced by:  sbcel1g 2016  sbceq1dig 2017  sbcel2g 2018  sbceq2dig 2019  csbidmg 2042  sbcbr12g 2668  sbcbr1g 2669  sbcbr2g 2670  csbfv2g 3749  csbopr12g 3993  csbopr1g 3994  csbopr2g 3995  fsumconst 7038  fsumcnlem 7986

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-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

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-sbc 1945  df-csb 2005

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