Theorem csbeq2i 2020

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 1946	. 2 |- (A e. D -> [A / x]B = C)
3		sbceqdig 2012	. 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 956   e. wcel 958  [wsbc 1170  [_csb 2001

This theorem is referenced by:  csbvarg 2021  csbopabg 2678

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-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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