Theorem 19.21v 1897

Description: Special case of Theorem 19.21 of [Margaris] p. 90. Notational convention: We sometimes suffix with "v" the label of a theorem eliminating a hypothesis such as ( ph -> A. x ph ) in 19.21 1607 via the use of distinct variable conditions combined with ax-17 1550. Conversely, we sometimes suffix with "f" the label of a theorem introducing such a hypothesis to eliminate the need for the distinct variable condition; e.g., euf 2060 derived from df-eu 2058. The "f" stands for "not free in" which is less restrictive than "does not occur in". (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
19.21v	|- ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) )

Distinct variable group:   ph, x

Allowed substitution hint:   ps( x)

Proof of Theorem 19.21v

Step	Hyp	Ref	Expression
1		ax-17 1550	. 2 |- ( ph -> A. x ph )
2	1	19.21h 1581	1 |- ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1371

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-gen 1473  ax-4 1534  ax-17 1550  ax-ial 1558  ax-i5r 1559

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  pm11.53  1920  cbval2  1946  cbvaldvaw  1955  sbhb  1969  2sb6  2013  sbcom2v  2014  2sb6rf  2019  2exsb  2038  moanim  2130  r3al  2552  ceqsralt  2804  rspc2gv  2896  euind  2967  reu2  2968  reuind  2985  unissb  3894  dfiin2g  3974  tfi  4648  asymref  5087  dff13  5860  mpo2eqb  6078