Theorem sbc19.20dv 1956

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

Hypothesis

Ref	Expression
sbc19.20dv.1	|- (ph -> (ps -> ch))

Assertion

Ref	Expression
sbc19.20dv	|- ((ph /\ A e. B) -> ([A / x]ps -> [A / x]ch))

Distinct variable group:   ph,x

Proof of Theorem sbc19.20dv

Step	Hyp	Ref	Expression
1		a4sbc 1916	. . . 4 |- (A e. B -> (A.x(ps -> ch) -> [A / x](ps -> ch)))
2		sbc19.20dv.1	. . . . 5 |- (ph -> (ps -> ch))
3	2	19.21aiv 1268	. . . 4 |- (ph -> A.x(ps -> ch))
4	1, 3	syl5 21	. . 3 |- (A e. B -> (ph -> [A / x](ps -> ch)))
5		sbcimg 1941	. . 3 |- (A e. B -> ([A / x](ps -> ch) <-> ([A / x]ps -> [A / x]ch)))
6	4, 5	sylibd 202	. 2 |- (A e. B -> (ph -> ([A / x]ps -> [A / x]ch)))
7	6	impcom 351	1 |- ((ph /\ A e. B) -> ([A / x]ps -> [A / x]ch))

Syntax hints:   -> wi 3   /\ wa 223  A.wal 950   e. wcel 1105  [wsbc 1153

This theorem is referenced by:  fsum1s 6898  fsump1s 6902

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-17 1190  ax-11o 1202  ax-ext 1436

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 957  df-sb 1155  df-clab 1441  df-cleq 1446  df-clel 1449  df-v 1787  df-sbc 1913

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