Theorem 19.36 2098

Description: Theorem 19.36 of [Margaris] p. 90. See 19.36v 1904 for a version requiring fewer axioms. (Contributed by NM, 24-Jun-1993.)

Hypothesis

Ref	Expression
19.36.1	|- F/ x ps

Assertion

Ref	Expression
19.36	|- ( E. x ( ph -> ps ) <-> ( A. x ph -> ps ) )

Proof of Theorem 19.36

Step	Hyp	Ref	Expression
1		19.35 1805	. 2 |- ( E. x ( ph -> ps ) <-> ( A. x ph -> E. x ps ) )
2		19.36.1	. . . 4 |- F/ x ps
3	2	19.9 2072	. . 3 |- ( E. x ps <-> ps )
4	3	imbi2i 326	. 2 |- ( ( A. x ph -> E. x ps ) <-> ( A. x ph -> ps ) )
5	1, 4	bitri 264	1 |- ( E. x ( ph -> ps ) <-> ( A. x ph -> ps ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   E.wex 1704   F/wnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-ex 1705  df-nf 1710

This theorem is referenced by:  19.36i  2099  19.12vv  2180  spcimgft  3284  19.12b  31707