Theorem sb8e 2425

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 2365	. 2 |- F/ x [ y / x ] ph
3		sbequ12 2111	. 2 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
4	1, 2, 3	cbvex 2272	1 |- ( E. x ph <-> E. y [ y / x ] ph )

Syntax hints:   <-> wb 196   E.wex 1704   F/wnf 1708   [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  sbnf2  2439  2sb8e  2467  sb8mo  2504  mo3  2507  bnj985  31023  bj-mo3OLD  32832  sbcexf  33918  exlimddvfi  33927  pm11.58  38590