Theorem sbcel2g 2015

Description: Move proper substitution in and out of a membership relation.

Assertion

Ref	Expression
sbcel2g	|- (A e. D -> ([A / x]B e. C <-> B e. [_A / x]_C))

Distinct variable group:   x,B

Proof of Theorem sbcel2g

Step	Hyp	Ref	Expression
1		sbcel12g 2011	. 2 |- (A e. D -> ([A / x]B e. C <-> [_A / x]_B e. [_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	eleq1d 1540	. 2 |- (A e. D -> ([_A / x]_B e. [_A / x]_C <-> B e. [_A / x]_C))
5	1, 4	bitrd 528	1 |- (A e. D -> ([A / x]B e. C <-> B e. [_A / x]_C))

Syntax hints:   -> wi 3   <-> wb 146   e. wcel 958  [wsbc 1170  [_csb 2001

This theorem is referenced by:  csbcomg 2017  sbccsbg 2022  hbcsb1gd 2027  hbcsbgd 2028

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-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

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-sbc 1942  df-csb 2002

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