Theorem sbccsb2 3761

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 3244	. 2 |- ( [. A / x ]. ph -> A e. _V )
2		elex 3021	. 2 |- ( A e. [_ A / x ]_ { x | ph } -> A e. _V )
3		abid 2439	. . . 4 |- ( x e. { x | ph } <-> ph )
4	3	sbcbii 3290	. . 3 |- ( [. A / x ]. x e. { x | ph } <-> [. A / x ]. ph )
5		sbcel12 3739	. . . 4 |- ( [. A / x ]. x e. { x | ph } <-> [_ A / x ]_ x e. [_ A / x ]_ { x | ph } )
6		csbvarg 3759	. . . . 5 |- ( A e. _V -> [_ A / x ]_ x = A )
7	6	eleq1d 2513	. . . 4 |- ( A e. _V -> ( [_ A / x ]_ x e. [_ A / x ]_ { x | ph } <-> A e. [_ A / x ]_ { x | ph } ) )
8	5, 7	syl5bb 265	. . 3 |- ( A e. _V -> ( [. A / x ]. x e. { x | ph } <-> A e. [_ A / x ]_ { x | ph } ) )
9	4, 8	syl5bbr 267	. 2 |- ( A e. _V -> ( [. A / x ]. ph <-> A e. [_ A / x ]_ { x | ph } ) )
10	1, 2, 9	pm5.21nii 359	1 |- ( [. A / x ]. ph <-> A e. [_ A / x ]_ { x | ph } )

Syntax hints:   <-> wb 189   e. wcel 1890   {cab 2437   _Vcvv 3012   [.wsbc 3234   [_csb 3330

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1672  ax-4 1685  ax-5 1761  ax-6 1808  ax-7 1854  ax-10 1918  ax-11 1923  ax-12 1936  ax-13 2091  ax-ext 2431

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1450  df-fal 1453  df-ex 1667  df-nf 1671  df-sb 1801  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-v 3014  df-sbc 3235  df-csb 3331  df-dif 3374  df-in 3378  df-ss 3385  df-nul 3699

This theorem is referenced by: (None)