Theorem sbexyz 1921

Description: Move existential quantifier in and out of substitution. Identical to sbex 1922 except that it has an additional distinct variable constraint on y and 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 1809	. . 3 |- ( [ z / y ] E. x ph <-> E. y ( y = z /\ E. x ph ) )
2		exdistr 1829	. . 3 |- ( E. y E. x ( y = z /\ ph ) <-> E. y ( y = z /\ E. x ph ) )
3		excom 1595	. . 3 |- ( E. y E. x ( y = z /\ ph ) <-> E. x E. y ( y = z /\ ph ) )
4	1, 2, 3	3bitr2i 206	. 2 |- ( [ z / y ] E. x ph <-> E. x E. y ( y = z /\ ph ) )
5		sb5 1809	. . 3 |- ( [ z / y ] ph <-> E. y ( y = z /\ ph ) )
6	5	exbii 1537	. 2 |- ( E. x [ z / y ] ph <-> E. x E. y ( y = z /\ ph ) )
7	4, 6	bitr4i 185	1 |- ( [ z / y ] E. x ph <-> E. x [ z / y ] ph )

Syntax hints:   /\ wa 102   <-> wb 103   E.wex 1422   [wsb 1686

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-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468

This theorem depends on definitions:  df-bi 115  df-sb 1687

This theorem is referenced by:  sbex  1922