Theorem sbcimg 2855

Description: Distribution of class substitution over implication. (Contributed by NM, 16-Jan-2004.)

Assertion

Ref	Expression
sbcimg	|- ( A e. V -> ( [. A / x ]. ( ph -> ps ) <-> ( [. A / x ]. ph -> [. A / x ]. ps ) ) )

Proof of Theorem sbcimg

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 2818	. 2 |- ( y = A -> ( [ y / x ] ( ph -> ps ) <-> [. A / x ]. ( ph -> ps ) ) )
2		dfsbcq2 2818	. . 3 |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
3		dfsbcq2 2818	. . 3 |- ( y = A -> ( [ y / x ] ps <-> [. A / x ]. ps ) )
4	2, 3	imbi12d 232	. 2 |- ( y = A -> ( ( [ y / x ] ph -> [ y / x ] ps ) <-> ( [. A / x ]. ph -> [. A / x ]. ps ) ) )
5		sbim 1868	. 2 |- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> [ y / x ] ps ) )
6	1, 4, 5	vtoclbg 2659	1 |- ( A e. V -> ( [. A / x ]. ( ph -> ps ) <-> ( [. A / x ]. ph -> [. A / x ]. ps ) ) )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1284   e. wcel 1433   [wsb 1685   [.wsbc 2815

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-sbc 2816

This theorem is referenced by:  sbcim1  2862  sbceqal  2869  sbc19.21g  2882  sbcssg  3350  iota4an  4906  sbcfung  4945  riotass2  5514