Theorem sb8eh 1848

Description: Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Proof rewritten by Jim Kingdon, 15-Jan-2018.)

Hypothesis

Ref	Expression
sb8eh.1	|- ( ph -> A. y ph )

Assertion

Ref	Expression
sb8eh	|- ( E. x ph <-> E. y [ y / x ] ph )

Proof of Theorem sb8eh

Step	Hyp	Ref	Expression
1		sb8eh.1	. 2 |- ( ph -> A. y ph )
2	1	hbsb3 1801	. 2 |- ( [ y / x ] ph -> A. x [ y / x ] ph )
3		sbequ12 1764	. 2 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
4	1, 2, 3	cbvexh 1748	1 |- ( E. x ph <-> E. y [ y / x ] ph )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1346   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:  exsb  2001