Theorem sbcel1g 2024

Description: Move proper substitution in and out of a membership relation. Note that the scope of [A / x] is the wff B e. C, whereas the scope of [_A / x]_ is the class B.

Assertion

Ref	Expression
sbcel1g	|- (A e. D -> ([A / x]B e. C <-> [_A / x]_B e. C))

Distinct variable group:   x,C

Proof of Theorem sbcel1g

Step	Hyp	Ref	Expression
1		sbcel12g 2022	. 2 |- (A e. D -> ([A / x]B e. C <-> [_A / x]_B e. [_A / x]_C))
2		ax-17 975	. . . 4 |- (y e. C -> A.x y e. C)
3	2	csbconstgf 2021	. . 3 |- (A e. D -> [_A / x]_C = C)
4	3	eleq2d 1548	. 2 |- (A e. D -> ([_A / x]_B e. [_A / x]_C <-> [_A / x]_B e. C))
5	1, 4	bitrd 531	1 |- (A e. D -> ([A / x]B e. C <-> [_A / x]_B e. C))

Syntax hints:   -> wi 3   <-> wb 146   e. wcel 962  [wsbc 1174  [_csb 2011

This theorem is referenced by:  ra4csbela 2053  fsum1s2 7042  csbfsum 7059  fsumrev 7061  fsumshf 7063

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-9 969  ax-10 970  ax-11 971  ax-12 972  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1127  ax-10o 1144  ax-16 1214  ax-11o 1222  ax-ext 1464

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 781  df-ex 985  df-sb 1176  df-clab 1470  df-cleq 1475  df-clel 1478  df-v 1819  df-sbc 1949  df-csb 2012

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