Theorem sbexyz 1996

Description: Move existential quantifier in and out of substitution. Identical to sbex 1997 except that it has an additional disjoint variable condition on y , z. (Contributed by Jim Kingdon, 29-Dec-2017.)

Assertion

Ref	Expression
sbexyz	|- ( [ z / y ] E. x ph <-> E. x [ z / y ] ph )

Distinct variable group:   x, y, z

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

Proof of Theorem sbexyz

Step	Hyp	Ref	Expression
1		sb5 1880	. . 3 |- ( [ z / y ] E. x ph <-> E. y ( y = z /\ E. x ph ) )
2		exdistr 1902	. . 3 |- ( E. y E. x ( y = z /\ ph ) <-> E. y ( y = z /\ E. x ph ) )
3		excom 1657	. . 3 |- ( E. y E. x ( y = z /\ ph ) <-> E. x E. y ( y = z /\ ph ) )
4	1, 2, 3	3bitr2i 207	. 2 |- ( [ z / y ] E. x ph <-> E. x E. y ( y = z /\ ph ) )
5		sb5 1880	. . 3 |- ( [ z / y ] ph <-> E. y ( y = z /\ ph ) )
6	5	exbii 1598	. 2 |- ( E. x [ z / y ] ph <-> E. x E. y ( y = z /\ ph ) )
7	4, 6	bitr4i 186	1 |- ( [ z / y ] E. x ph <-> E. x [ z / y ] ph )

Syntax hints:   /\ wa 103   <-> wb 104   E.wex 1485   [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-7 1441  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:  sbex  1997