Theorem sbcimdv 2817

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 1751	. . . 4 |- (ph -> A.x(ps -> ch))
3		spsbc 2769	. . . 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 2798	. . 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 1240   e. wcel 1390   [.wsbc 2758

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759

This theorem is referenced by: (None)