Theorem sbccsb2 4005

Description: Substitution into a wff expressed in using substitution into a class. (Contributed by NM, 27-Nov-2005.) (Revised by NM, 18-Aug-2018.)

Assertion

Ref	Expression
sbccsb2	|- ( [. A / x ]. ph <-> A e. [_ A / x ]_ { x | ph } )

Proof of Theorem sbccsb2

Step	Hyp	Ref	Expression
1		sbcex 3445	. 2 |- ( [. A / x ]. ph -> A e. _V )
2		elex 3212	. 2 |- ( A e. [_ A / x ]_ { x | ph } -> A e. _V )
3		abid 2610	. . . 4 |- ( x e. { x | ph } <-> ph )
4	3	sbcbii 3491	. . 3 |- ( [. A / x ]. x e. { x | ph } <-> [. A / x ]. ph )
5		sbcel12 3983	. . . 4 |- ( [. A / x ]. x e. { x | ph } <-> [_ A / x ]_ x e. [_ A / x ]_ { x | ph } )
6		csbvarg 4003	. . . . 5 |- ( A e. _V -> [_ A / x ]_ x = A )
7	6	eleq1d 2686	. . . 4 |- ( A e. _V -> ( [_ A / x ]_ x e. [_ A / x ]_ { x | ph } <-> A e. [_ A / x ]_ { x | ph } ) )
8	5, 7	syl5bb 272	. . 3 |- ( A e. _V -> ( [. A / x ]. x e. { x | ph } <-> A e. [_ A / x ]_ { x | ph } ) )
9	4, 8	syl5bbr 274	. 2 |- ( A e. _V -> ( [. A / x ]. ph <-> A e. [_ A / x ]_ { x | ph } ) )
10	1, 2, 9	pm5.21nii 368	1 |- ( [. A / x ]. ph <-> A e. [_ A / x ]_ { x | ph } )

Syntax hints:   <-> wb 196   e. wcel 1990   {cab 2608   _Vcvv 3200   [.wsbc 3435   [_csb 3533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-nul 3916

This theorem is referenced by: (None)