Theorem sb8eh 1776

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 1729	. 2 |- ( [ y / x ] ph -> A. x [ y / x ] ph )
3		sbequ12 1694	. 2 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
4	1, 2, 3	cbvexh 1678	1 |- ( E. x ph <-> E. y [ y / x ] ph )

Syntax hints:   -> wi 4   <-> wb 103   A.wal 1282   E.wex 1421   [wsb 1685

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 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-sb 1686

This theorem is referenced by:  exsb  1925