Theorem spvw 1666

Description: Version of sp 1747 when x does not occur in ph. This provides the other direction of ax-17 1616. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.) (Proof shortened by Wolf Lammen, 4-Dec-2017.)

Assertion

Ref	Expression
spvw	|- (A.xph -> ph)

Distinct variable group:   ph,x

Proof of Theorem spvw

Step	Hyp	Ref	Expression
1		19.3v 1665	. 2 |- (A.xph <-> ph)
2	1	biimpi 186	1 |- (A.xph -> ph)

Syntax hints:   -> wi 4  A.wal 1540

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

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542

This theorem is referenced by:  19.3vOLD  1696