Theorem sb8eh 1866

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 1819	. 2 |- ( [ y / x ] ph -> A. x [ y / x ] ph )
3		sbequ12 1782	. 2 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
4	1, 2, 3	cbvexh 1766	1 |- ( E. x ph <-> E. y [ y / x ] ph )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1362   E.wex 1503   [wsb 1773

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-sb 1774

This theorem is referenced by:  exsb  2024