Theorem sbcimdv 3108

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 1631	. . . 4 |- (ph -> A.x(ps -> ch))
3		spsbc 3059	. . . 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 3088	. . 3 |- (A e. V -> ( [.A / x ].(ps -> ch) <-> ( [.A / x ].ps -> [.A / x ].ch)))
6	4, 5	sylibd 205	. 2 |- (A e. V -> (ph -> ( [.A / x ].ps -> [.A / x ].ch)))
7	6	impcom 419	1 |- ((ph /\ A e. V) -> ( [.A / x ].ps -> [.A / x ].ch))

Syntax hints:   -> wi 4   /\ wa 358  A.wal 1540   e. wcel 1710   [.wsbc 3047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sbc 3048

This theorem is referenced by: (None)