Theorem sbcel2g 2011

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 2007	. 2 |- (A e. D -> ([A / x]B e. C <-> [_A / x]_B e. [_A / x]_C))
2		ax-17 969	. . . 4 |- (y e. B -> A.x y e. B)
3	2	csbconstgf 2006	. . 3 |- (A e. D -> [_A / x]_B = B)
4	3	eleq1d 1537	. 2 |- (A e. D -> ([_A / x]_B e. [_A / x]_C <-> B e. [_A / x]_C))
5	1, 4	bitrd 527	1 |- (A e. D -> ([A / x]B e. C <-> B e. [_A / x]_C))

Syntax hints:   -> wi 3   <-> wb 146   e. wcel 956  [wsbc 1168  [_csb 1997

This theorem is referenced by:  csbcomg 2013  sbccsbg 2018  hbcsb1gd 2023  hbcsbgd 2024

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-sbc 1938  df-csb 1998

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