Theorem sborv 1811

Description: Version of sbor 1869 where x and y are distinct. (Contributed by Jim Kingdon, 3-Feb-2018.)

Assertion

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

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)   ps( x, y)

Proof of Theorem sborv

Step	Hyp	Ref	Expression
1		sb5 1808	. . 3 |- ( [ y / x ] ( ph \/ ps ) <-> E. x ( x = y /\ ( ph \/ ps ) ) )
2		andi 764	. . . 4 |- ( ( x = y /\ ( ph \/ ps ) ) <-> ( ( x = y /\ ph ) \/ ( x = y /\ ps ) ) )
3	2	exbii 1536	. . 3 |- ( E. x ( x = y /\ ( ph \/ ps ) ) <-> E. x ( ( x = y /\ ph ) \/ ( x = y /\ ps ) ) )
4		19.43 1559	. . 3 |- ( E. x ( ( x = y /\ ph ) \/ ( x = y /\ ps ) ) <-> ( E. x ( x = y /\ ph ) \/ E. x ( x = y /\ ps ) ) )
5	1, 3, 4	3bitri 204	. 2 |- ( [ y / x ] ( ph \/ ps ) <-> ( E. x ( x = y /\ ph ) \/ E. x ( x = y /\ ps ) ) )
6		sb5 1808	. . 3 |- ( [ y / x ] ph <-> E. x ( x = y /\ ph ) )
7		sb5 1808	. . 3 |- ( [ y / x ] ps <-> E. x ( x = y /\ ps ) )
8	6, 7	orbi12i 713	. 2 |- ( ( [ y / x ] ph \/ [ y / x ] ps ) <-> ( E. x ( x = y /\ ph ) \/ E. x ( x = y /\ ps ) ) )
9	5, 8	bitr4i 185	1 |- ( [ y / x ] ( ph \/ ps ) <-> ( [ y / x ] ph \/ [ y / x ] ps ) )

Syntax hints:   /\ wa 102   <-> wb 103   \/ wo 661   E.wex 1421   [wsb 1685

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-sb 1686

This theorem is referenced by:  sbor  1869