Theorem sb8e 1855

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
sb8e.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		sb8e.1	. 2 |- F/ y ph
2	1	nfs1 1807	. 2 |- F/ x [ y / x ] ph
3		sbequ12 1769	. 2 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
4	1, 2, 3	cbvex 1754	1 |- ( E. x ph <-> E. y [ y / x ] ph )

Syntax hints:   <-> wb 105   F/wnf 1458   E.wex 1490   [wsb 1760

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-11 1504  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532

This theorem depends on definitions:  df-bi 117  df-nf 1459  df-sb 1761

This theorem is referenced by:  sb8mo  2038  mo2n  2052  mor  2066  nfrexdya  2511