Theorem sbccsb2 3814

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 3304	. 2 |- ( [. A / x ]. ph -> A e. _V )
2		elex 3087	. 2 |- ( A e. [_ A / x ]_ { x | ph } -> A e. _V )
3		abid 2441	. . . 4 |- ( x e. { x | ph } <-> ph )
4	3	sbcbii 3354	. . 3 |- ( [. A / x ]. x e. { x | ph } <-> [. A / x ]. ph )
5		sbcel12 3786	. . . 4 |- ( [. A / x ]. x e. { x | ph } <-> [_ A / x ]_ x e. [_ A / x ]_ { x | ph } )
6		csbvarg 3811	. . . . 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 353	1 |- ( [. A / x ]. ph <-> A e. [_ A / x ]_ { x | ph } )

Syntax hints:   <-> wb 184   e. wcel 1758   {cab 2439   _Vcvv 3078   [.wsbc 3294   [_csb 3398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-in 3446  df-ss 3453  df-nul 3749

This theorem is referenced by: (None)