Theorem sbc19.21g 1470

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		sbcim 1460	. 2 |- (A e. B -> ([A / x](ph -> ps) <-> ([A / x]ph -> [A / x]ps)))
2		sbc19.21g.1	. . . . 5 |- (ph -> A.xph)
3	2	sbcgf 1469	. . . 4 |- (A e. B -> ([A / x]ph <-> ph))
4	3	imbi1d 465	. . 3 |- (A e. B -> (([A / x]ph -> [A / x]ps) <-> (ph -> [A / x]ps)))
5	4	bicomd 399	. 2 |- (A e. B -> ((ph -> [A / x]ps) <-> ([A / x]ph -> [A / x]ps)))
6	1, 5	bitr4d 409	1 |- (A e. B -> ([A / x](ph -> ps) <-> (ph -> [A / x]ps)))

Syntax hints:   -> wi 2   <-> wb 127  A.wal 672   e. wcel 1092  [wsbc 1440

This theorem is referenced by:  nn1suc 4435

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-sbc 1441

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