Theorem sbexyz 2031

Description: Move existential quantifier in and out of substitution. Identical to sbex 2032 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 1911	. . 3 |- ( [ z / y ] E. x ph <-> E. y ( y = z /\ E. x ph ) )
2		exdistr 1933	. . 3 |- ( E. y E. x ( y = z /\ ph ) <-> E. y ( y = z /\ E. x ph ) )
3		excom 1687	. . 3 |- ( E. y E. x ( y = z /\ ph ) <-> E. x E. y ( y = z /\ ph ) )
4	1, 2, 3	3bitr2i 208	. 2 |- ( [ z / y ] E. x ph <-> E. x E. y ( y = z /\ ph ) )
5		sb5 1911	. . 3 |- ( [ z / y ] ph <-> E. y ( y = z /\ ph ) )
6	5	exbii 1628	. 2 |- ( E. x [ z / y ] ph <-> E. x E. y ( y = z /\ ph ) )
7	4, 6	bitr4i 187	1 |- ( [ z / y ] E. x ph <-> E. x [ z / y ] ph )

Syntax hints:   /\ wa 104   <-> wb 105   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-5 1470  ax-7 1471  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:  sbex  2032