Theorem sb8e 2169

Description: Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)

Hypothesis

Ref	Expression
sb5rf.1	|- F/ y ph

Assertion

Ref	Expression
sb8e	|- ( E. x ph <-> E. y [ y / x ] ph )

Proof of Theorem sb8e

Step	Hyp	Ref	Expression
1		sb5rf.1	. 2 |- F/ y ph
2	1	nfs1 2105	. 2 |- F/ x [ y / x ] ph
3		sbequ12 1993	. 2 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
4	1, 2, 3	cbvex 2023	1 |- ( E. x ph <-> E. y [ y / x ] ph )

Syntax hints:   <-> wb 184   E.wex 1613   F/wnf 1617   [wsb 1740

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000

This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1614  df-nf 1618  df-sb 1741

This theorem is referenced by:  sbnf2  2184  2sb8e  2212  sb8mo  2321  mo3  2324  mo3OLD  2325  mopickOLD  2357  sbcexf  30766  exlimddvfi  30775  pm11.58  31543  bnj985  34197