Theorem csbeq2i 2016

Description: Formula-building inference rule for class substitution.

Hypothesis

Ref	Expression
csbeq2i.1	|- B = C

Assertion

Ref	Expression
csbeq2i	|- (A e. D -> [_A / x]_B = [_A / x]_C)

Proof of Theorem csbeq2i

Step	Hyp	Ref	Expression
1		csbeq2i.1	. . 3 |- B = C
2	1	sbcth 1942	. 2 |- (A e. D -> [A / x]B = C)
3		sbceqdig 2008	. 2 |- (A e. D -> ([A / x]B = C <-> [_A / x]_B = [_A / x]_C))
4	2, 3	mpbid 195	1 |- (A e. D -> [_A / x]_B = [_A / x]_C)

Syntax hints:   -> wi 3   = wceq 954   e. wcel 956  [wsbc 1168  [_csb 1997

This theorem is referenced by:  csbvarg 2017  csbopabg 2673

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-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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