Theorem sbcied 3472

Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.)

Hypotheses

Ref	Expression
sbcied.1	|- ( ph -> A e. V )
sbcied.2	|- ( ( ph /\ x = A ) -> ( ps <-> ch ) )

Assertion

Ref	Expression
sbcied	|- ( ph -> ( [. A / x ]. ps <-> ch ) )

Distinct variable groups:   x, A   ph, x   ch, x

Allowed substitution hints:   ps( x)   V( x)

Proof of Theorem sbcied

Step	Hyp	Ref	Expression
1		sbcied.1	. 2 |- ( ph -> A e. V )
2		sbcied.2	. 2 |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )
3		nfv 1843	. 2 |- F/ x ph
4		nfvd 1844	. 2 |- ( ph -> F/ x ch )
5	1, 2, 3, 4	sbciedf 3471	1 |- ( ph -> ( [. A / x ]. ps <-> ch ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   [.wsbc 3435

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-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202  df-sbc 3436

This theorem is referenced by:  sbcied2  3473  sbc2iedv  3506  sbc3ie  3507  sbcralt  3510  euotd  4975  fmptsnd  6435  riota5f  6636  fpwwe2lem12  9463  fpwwe2lem13  9464  brfi1uzind  13280  opfi1uzind  13283  brfi1uzindOLD  13286  opfi1uzindOLD  13289  sbcie3s  15917  issubc  16495  gsumvalx  17270  dmdprd  18397  dprdval  18402  issrg  18507  issrng  18850  islmhm  19027  isassa  19315  isphl  19973  istmd  21878  istgp  21881  isnlm  22479  isclm  22864  iscph  22970  iscms  23142  limcfval  23636  ewlksfval  26497  sbcies  29322  abfmpeld  29454  abfmpel  29455  isomnd  29701  isorng  29799