Theorem sbccsb2 3844

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 3334	. 2 |- ( [. A / x ]. ph -> A e. _V )
2		elex 3115	. 2 |- ( A e. [_ A / x ]_ { x | ph } -> A e. _V )
3		abid 2441	. . . 4 |- ( x e. { x | ph } <-> ph )
4	3	sbcbii 3380	. . 3 |- ( [. A / x ]. x e. { x | ph } <-> [. A / x ]. ph )
5		sbcel12 3822	. . . 4 |- ( [. A / x ]. x e. { x | ph } <-> [_ A / x ]_ x e. [_ A / x ]_ { x | ph } )
6		csbvarg 3842	. . . . 5 |- ( A e. _V -> [_ A / x ]_ x = A )
7	6	eleq1d 2523	. . . 4 |- ( A e. _V -> ( [_ A / x ]_ x e. [_ A / x ]_ { x | ph } <-> A e. [_ A / x ]_ { x | ph } ) )
8	5, 7	syl5bb 257	. . 3 |- ( A e. _V -> ( [. A / x ]. x e. { x | ph } <-> A e. [_ A / x ]_ { x | ph } ) )
9	4, 8	syl5bbr 259	. 2 |- ( A e. _V -> ( [. A / x ]. ph <-> A e. [_ A / x ]_ { x | ph } ) )
10	1, 2, 9	pm5.21nii 351	1 |- ( [. A / x ]. ph <-> A e. [_ A / x ]_ { x | ph } )

Syntax hints:   <-> wb 184   e. wcel 1823   {cab 2439   _Vcvv 3106   [.wsbc 3324   [_csb 3420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-in 3468  df-ss 3475  df-nul 3784

This theorem is referenced by: (None)