Theorem sborv 1914

Description: Version of sbor 1982 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 1911	. . 3 |- ( [ y / x ] ( ph \/ ps ) <-> E. x ( x = y /\ ( ph \/ ps ) ) )
2		andi 820	. . . 4 |- ( ( x = y /\ ( ph \/ ps ) ) <-> ( ( x = y /\ ph ) \/ ( x = y /\ ps ) ) )
3	2	exbii 1628	. . 3 |- ( E. x ( x = y /\ ( ph \/ ps ) ) <-> E. x ( ( x = y /\ ph ) \/ ( x = y /\ ps ) ) )
4		19.43 1651	. . 3 |- ( E. x ( ( x = y /\ ph ) \/ ( x = y /\ ps ) ) <-> ( E. x ( x = y /\ ph ) \/ E. x ( x = y /\ ps ) ) )
5	1, 3, 4	3bitri 206	. 2 |- ( [ y / x ] ( ph \/ ps ) <-> ( E. x ( x = y /\ ph ) \/ E. x ( x = y /\ ps ) ) )
6		sb5 1911	. . 3 |- ( [ y / x ] ph <-> E. x ( x = y /\ ph ) )
7		sb5 1911	. . 3 |- ( [ y / x ] ps <-> E. x ( x = y /\ ps ) )
8	6, 7	orbi12i 766	. 2 |- ( ( [ y / x ] ph \/ [ y / x ] ps ) <-> ( E. x ( x = y /\ ph ) \/ E. x ( x = y /\ ps ) ) )
9	5, 8	bitr4i 187	1 |- ( [ y / x ] ( ph \/ ps ) <-> ( [ y / x ] ph \/ [ y / x ] ps ) )

Syntax hints:   /\ wa 104   <-> wb 105   \/ wo 710   E.wex 1515   [wsb 1785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557

This theorem depends on definitions:  df-bi 117  df-sb 1786

This theorem is referenced by:  sbor  1982