Theorem sbor 887

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 886	. . 3 |- ([y / x](-. ph -> ps) <-> ([y / x] -. ph -> [y / x]ps))
2		sbn 882	. . . 4 |- ([y / x] -. ph <-> -. [y / x]ph)
3	2	imbi1i 161	. . 3 |- (([y / x] -. ph -> [y / x]ps) <-> (-. [y / x]ph -> [y / x]ps))
4	1, 3	bitr 151	. 2 |- ([y / x](-. ph -> ps) <-> (-. [y / x]ph -> [y / x]ps))
5		df-or 197	. . 3 |- ((ph \/ ps) <-> (-. ph -> ps))
6	5	bisb 855	. 2 |- ([y / x](ph \/ ps) <-> [y / x](-. ph -> ps))
7		df-or 197	. 2 |- (([y / x]ph \/ [y / x]ps) <-> (-. [y / x]ph -> [y / x]ps))
8	4, 6, 7	3bitr4 158	1 |- ([y / x](ph \/ ps) <-> ([y / x]ph \/ [y / x]ps))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   \/ wo 195  [wsb 852

This theorem is referenced by:  sbcor 1462  unab 1691

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

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853

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