Theorem sb8e 1737

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 1690	. 2 |- F/ x [ y / x ] ph
3		sbequ12 1654	. 2 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
4	1, 2, 3	cbvex 1639	1 |- ( E. x ph <-> E. y [ y / x ] ph )

Syntax hints:   <-> wb 98   F/wnf 1349   E.wex 1381   [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646

This theorem is referenced by:  sb8mo  1914  mo2n  1928  mor  1942  nfrexdya  2359