Theorem spsbe 1865

Description: A specialization theorem, mostly the same as Theorem 19.8 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 29-Dec-2017.)

Assertion

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

Proof of Theorem spsbe

Step	Hyp	Ref	Expression
1		sb1 1789	. 2 |- ( [ y / x ] ph -> E. x ( x = y /\ ph ) )
2		simpr 110	. . 3 |- ( ( x = y /\ ph ) -> ph )
3	2	eximi 1623	. 2 |- ( E. x ( x = y /\ ph ) -> E. x ph )
4	1, 3	syl 14	1 |- ( [ y / x ] ph -> E. x ph )

Syntax hints:   -> wi 4   /\ wa 104   E.wex 1515   [wsb 1785

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 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533  ax-ial 1557

This theorem depends on definitions:  df-bi 117  df-sb 1786

This theorem is referenced by:  sbft  1871