Theorem speiv 2000

Description: Inference from existential specialization, using implicit substitution. (Contributed by NM, 19-Aug-1993.)

Hypotheses

Ref	Expression
speiv.1	|- (x = y -> (ph <-> ps))
speiv.2	|- ps

Assertion

Ref	Expression
speiv	|- E.xph

Distinct variable group:   ps,x

Allowed substitution hints:   ph(x,y)   ps(y)

Proof of Theorem speiv

Step	Hyp	Ref	Expression
1		speiv.2	. 2 |- ps
2		speiv.1	. . . 4 |- (x = y -> (ph <-> ps))
3	2	biimprd 214	. . 3 |- (x = y -> (ps -> ph))
4	3	spimev 1999	. 2 |- (ps -> E.xph)
5	1, 4	ax-mp 5	1 |- E.xph

Syntax hints:   -> wi 4   <-> wb 176  E.wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  ce0addcnnul  6180