Theorem sbccsb2g 2026

Description: Substitution into a wff expressed in using substitution into a class.

Assertion

Ref	Expression
sbccsb2g	|- (A e. B -> ([A / x]ph <-> A e. [_A / x]_{x | ph}))

Proof of Theorem sbccsb2g

Step	Hyp	Ref	Expression
1		sbcel12g 2014	. 2 |- (A e. B -> ([A / x]x e. {x | ph} <-> [_A / x]_x e. [_A / x]_{x | ph}))
2		abid 1468	. . 3 |- (x e. {x | ph} <-> ph)
3	2	sbcbii 1981	. 2 |- (A e. B -> ([A / x]x e. {x | ph} <-> [A / x]ph))
4		csbvarg 2024	. . 3 |- (A e. B -> [_A / x]_x = A)
5	4	eleq1d 1543	. 2 |- (A e. B -> ([_A / x]_x e. [_A / x]_{x | ph} <-> A e. [_A / x]_{x | ph}))
6	1, 3, 5	3bitr3d 550	1 |- (A e. B -> ([A / x]ph <-> A e. [_A / x]_{x | ph}))

Syntax hints:   -> wi 3   <-> wb 146   e. wcel 960  [wsbc 1172  {cab 1466  [_csb 2004

This theorem is referenced by:  sbcnestg 2041

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-sbc 1945  df-csb 2005

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