Theorem sbc19.21g 1983

Description: Substitution for a variable not free in antecedent affects only the consequent.

Hypothesis

Ref	Expression
sbc19.21g.1	|- (ph -> A.xph)

Assertion

Ref	Expression
sbc19.21g	|- (A e. B -> ([A / x](ph -> ps) <-> (ph -> [A / x]ps)))

Proof of Theorem sbc19.21g

Step	Hyp	Ref	Expression
1		sbcimg 1966	. 2 |- (A e. B -> ([A / x](ph -> ps) <-> ([A / x]ph -> [A / x]ps)))
2		sbc19.21g.1	. . . 4 |- (ph -> A.xph)
3	2	sbcgf 1982	. . 3 |- (A e. B -> ([A / x]ph <-> ph))
4	3	imbi1d 612	. 2 |- (A e. B -> (([A / x]ph -> [A / x]ps) <-> (ph -> [A / x]ps)))
5	1, 4	bitrd 527	1 |- (A e. B -> ([A / x](ph -> ps) <-> (ph -> [A / x]ps)))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 952   e. wcel 956  [wsbc 1168

This theorem is referenced by:  nn1suc 5895

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-sbc 1938

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