Theorem sbcimg 2804

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 2767	. 2 |- ( y = A -> ( [ y / x ] ( ph -> ps ) <-> [. A / x ]. ( ph -> ps ) ) )
2		dfsbcq2 2767	. . 3 |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
3		dfsbcq2 2767	. . 3 |- ( y = A -> ( [ y / x ] ps <-> [. A / x ]. ps ) )
4	2, 3	imbi12d 223	. 2 |- ( y = A -> ( ( [ y / x ] ph -> [ y / x ] ps ) <-> ( [. A / x ]. ph -> [. A / x ]. ps ) ) )
5		sbim 1827	. 2 |- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> [ y / x ] ps ) )
6	1, 4, 5	vtoclbg 2614	1 |- ( A e. V -> ( [. A / x ]. ( ph -> ps ) <-> ( [. A / x ]. ph -> [. A / x ]. ps ) ) )

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1243   e. wcel 1393   [wsb 1645   [.wsbc 2764

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765

This theorem is referenced by:  sbceqal  2814  sbcimdv  2823  sbc19.21g  2826  sbcssg  3330  iota4an  4886  sbcfung  4925  riotass2  5494