Theorem sbccsb2g 2023

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 2011	. 2 |- (A e. B -> ([A / x]x e. {x | ph} <-> [_A / x]_x e. [_A / x]_{x | ph}))
2		abid 1465	. . 3 |- (x e. {x | ph} <-> ph)
3	2	sbcbii 1978	. 2 |- (A e. B -> ([A / x]x e. {x | ph} <-> [A / x]ph))
4		csbvarg 2021	. . 3 |- (A e. B -> [_A / x]_x = A)
5	4	eleq1d 1540	. 2 |- (A e. B -> ([_A / x]_x e. [_A / x]_{x | ph} <-> A e. [_A / x]_{x | ph}))
6	1, 3, 5	3bitr3d 548	1 |- (A e. B -> ([A / x]ph <-> A e. [_A / x]_{x | ph}))

Syntax hints:   -> wi 3   <-> wb 146   e. wcel 958  [wsbc 1170  {cab 1463  [_csb 2001

This theorem is referenced by:  sbcnestg 2038

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-11 967  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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