Theorem sbc19.20dv 1985

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 1945	. . . 4 |- (A e. B -> (A.x(ps -> ch) -> [A / x](ps -> ch)))
2		sbc19.20dv.1	. . . . 5 |- (ph -> (ps -> ch))
3	2	19.21aiv 1286	. . . 4 |- (ph -> A.x(ps -> ch))
4	1, 3	syl5 21	. . 3 |- (A e. B -> (ph -> [A / x](ps -> ch)))
5		sbcimg 1970	. . 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 954   e. wcel 958  [wsbc 1170

This theorem is referenced by:  fsum1s 7009  fsump1s 7013

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-sbc 1942

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