Theorem orbi2d 682

Description: Deduction adding a left disjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
bid.1	|- (ph -> (ps <-> ch))

Assertion

Ref	Expression
orbi2d	|- (ph -> ((th \/ ps) <-> (th \/ ch)))

Proof of Theorem orbi2d

Step	Hyp	Ref	Expression
1		bid.1	. . 3 |- (ph -> (ps <-> ch))
2	1	imbi2d 307	. 2 |- (ph -> ((-. th -> ps) <-> (-. th -> ch)))
3		df-or 359	. 2 |- ((th \/ ps) <-> (-. th -> ps))
4		df-or 359	. 2 |- ((th \/ ch) <-> (-. th -> ch))
5	2, 3, 4	3bitr4g 279	1 |- (ph -> ((th \/ ps) <-> (th \/ ch)))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   \/ wo 357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-or 359

This theorem is referenced by:  orbi1d  683  orbi12d  690  cad1  1398  eueq2  3011  sbc2or  3055  rexprg  3777  rextpg  3779  clos1basesucg  5885  nc0suc  6218  nmembers1lem3  6271