Theorem sbor 1234

Description: Logical OR inside and outside of substitution are equivalent.

Assertion

Ref	Expression
sbor	|- ([y / x](ph \/ ps) <-> ([y / x]ph \/ [y / x]ps))

Proof of Theorem sbor

Step	Hyp	Ref	Expression
1		sbim 1233	. . 3 |- ([y / x](-. ph -> ps) <-> ([y / x] -. ph -> [y / x]ps))
2		sbn 1230	. . . 4 |- ([y / x] -. ph <-> -. [y / x]ph)
3	2	imbi1i 186	. . 3 |- (([y / x] -. ph -> [y / x]ps) <-> (-. [y / x]ph -> [y / x]ps))
4	1, 3	bitr 173	. 2 |- ([y / x](-. ph -> ps) <-> (-. [y / x]ph -> [y / x]ps))
5		df-or 224	. . 3 |- ((ph \/ ps) <-> (-. ph -> ps))
6	5	sbbii 1173	. 2 |- ([y / x](ph \/ ps) <-> [y / x](-. ph -> ps))
7		df-or 224	. 2 |- (([y / x]ph \/ [y / x]ps) <-> (-. [y / x]ph -> [y / x]ps))
8	4, 6, 7	3bitr4 183	1 |- ([y / x](ph \/ ps) <-> ([y / x]ph \/ [y / x]ps))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222  [wsbc 1169

This theorem is referenced by:  sbcorg 1969  unab 2264

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-10 965  ax-12 967  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-11o 1217

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171

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