Theorem sb8e 2256

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 2196	. 2 |- F/ x [ y / x ] ph
3		sbequ12 2085	. 2 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
4	1, 2, 3	cbvex 2117	1 |- ( E. x ph <-> E. y [ y / x ] ph )

Syntax hints:   <-> wb 188   E.wex 1665   F/wnf 1669   [wsb 1799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093

This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1666  df-nf 1670  df-sb 1800

This theorem is referenced by:  sbnf2  2270  2sb8e  2298  sb8mo  2335  mo3  2338  bnj985  29776  bj-mo3OLD  31459  sbcexf  32365  exlimddvfi  32374  pm11.58  36751