Theorem sbcimdv 2823

Description: Substitution analog of Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 11-Nov-2005.)

Hypothesis

Ref	Expression
sbcimdv.1	|- ( ph -> ( ps -> ch ) )

Assertion

Ref	Expression
sbcimdv	|- ( ( ph /\ A e. V ) -> ( [. A / x ]. ps -> [. A / x ]. ch ) )

Distinct variable group:   ph, x

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

Proof of Theorem sbcimdv

Step	Hyp	Ref	Expression
1		sbcimdv.1	. . . . 5 |- ( ph -> ( ps -> ch ) )
2	1	alrimiv 1754	. . . 4 |- ( ph -> A. x ( ps -> ch ) )
3		spsbc 2775	. . . 4 |- ( A e. V -> ( A. x ( ps -> ch ) -> [. A / x ]. ( ps -> ch ) ) )
4	2, 3	syl5 28	. . 3 |- ( A e. V -> ( ph -> [. A / x ]. ( ps -> ch ) ) )
5		sbcimg 2804	. . 3 |- ( A e. V -> ( [. A / x ]. ( ps -> ch ) <-> ( [. A / x ]. ps -> [. A / x ]. ch ) ) )
6	4, 5	sylibd 138	. 2 |- ( A e. V -> ( ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) )
7	6	impcom 116	1 |- ( ( ph /\ A e. V ) -> ( [. A / x ]. ps -> [. A / x ]. ch ) )

Syntax hints:   -> wi 4   /\ wa 97   A.wal 1241   e. wcel 1393   [.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: (None)