Theorem sbanv 1882

Description: Version of sban 1948 where x and y are distinct. (Contributed by Jim Kingdon, 24-Dec-2017.)

Assertion

Ref	Expression
sbanv	|- ( [ 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 sbanv

Step	Hyp	Ref	Expression
1		sb6 1879	. 2 |- ( [ y / x ] ( ph /\ ps ) <-> A. x ( x = y -> ( ph /\ ps ) ) )
2		sb6 1879	. . . 4 |- ( [ y / x ] ph <-> A. x ( x = y -> ph ) )
3		sb6 1879	. . . 4 |- ( [ y / x ] ps <-> A. x ( x = y -> ps ) )
4	2, 3	anbi12i 457	. . 3 |- ( ( [ y / x ] ph /\ [ y / x ] ps ) <-> ( A. x ( x = y -> ph ) /\ A. x ( x = y -> ps ) ) )
5		19.26 1474	. . 3 |- ( A. x ( ( x = y -> ph ) /\ ( x = y -> ps ) ) <-> ( A. x ( x = y -> ph ) /\ A. x ( x = y -> ps ) ) )
6		pm4.76 599	. . . 4 |- ( ( ( x = y -> ph ) /\ ( x = y -> ps ) ) <-> ( x = y -> ( ph /\ ps ) ) )
7	6	albii 1463	. . 3 |- ( A. x ( ( x = y -> ph ) /\ ( x = y -> ps ) ) <-> A. x ( x = y -> ( ph /\ ps ) ) )
8	4, 5, 7	3bitr2i 207	. 2 |- ( ( [ y / x ] ph /\ [ y / x ] ps ) <-> A. x ( x = y -> ( ph /\ ps ) ) )
9	1, 8	bitr4i 186	1 |- ( [ y / x ] ( ph /\ ps ) <-> ( [ y / x ] ph /\ [ y / x ] ps ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1346   [wsb 1755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-sb 1756

This theorem is referenced by:  sban  1948