Theorem sbcimdv 2519

Description: Substitution analog of Theorem 19.20 of [Margaris] p. 90.

Hypothesis

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

Assertion

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

Distinct variable group:   ph,x

Proof of Theorem sbcimdv

Step	Hyp	Ref	Expression
1		a4sbc 2457	. . . 4 |- (A e. B -> (A.x(ps -> ch) -> [A / x](ps -> ch)))
2		sbcimdv.1	. . . . 5 |- (ph -> (ps -> ch))
3	2	19.21aiv 1664	. . . 4 |- (ph -> A.x(ps -> ch))
4	1, 3	syl5 20	. . 3 |- (A e. B -> (ph -> [A / x](ps -> ch)))
5		sbcimg 2496	. . 3 |- (A e. B -> ([A / x](ps -> ch) <-> ([A / x]ps -> [A / x]ch)))
6	4, 5	sylibd 219	. 2 |- (A e. B -> (ph -> ([A / x]ps -> [A / x]ch)))
7	6	impcom 378	1 |- ((ph /\ A e. B) -> ([A / x]ps -> [A / x]ch))

Syntax hints:   -> wi 3   /\ wa 240  A.wal 1296   e. wcel 1300  [wsbc 1534

This theorem is referenced by:  fsum1s 8269  fsump1s 8273  fprod1s 14677  fprodp1s 14682

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-sbc 2454

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