Theorem 19.36-1 1603

Description: Closed form of 19.36i 1602. One direction of Theorem 19.36 of [Margaris] p. 90. The converse holds in classical logic, but does not hold (for all propositions) in intuitionistic logic. (Contributed by Jim Kingdon, 20-Jun-2018.)

Hypothesis

Ref	Expression
19.36-1.1	|- F/ x ps

Assertion

Ref	Expression
19.36-1	|- ( E. x ( ph -> ps ) -> ( A. x ph -> ps ) )

Proof of Theorem 19.36-1

Step	Hyp	Ref	Expression
1		19.35-1 1555	. 2 |- ( E. x ( ph -> ps ) -> ( A. x ph -> E. x ps ) )
2		19.36-1.1	. . 3 |- F/ x ps
3	2	19.9 1575	. 2 |- ( E. x ps <-> ps )
4	1, 3	syl6ib 159	1 |- ( E. x ( ph -> ps ) -> ( A. x ph -> ps ) )

Syntax hints:   -> wi 4   A.wal 1282   F/wnf 1389   E.wex 1421

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-nf 1390

This theorem is referenced by:  vtocl2  2654  vtocl3  2655  spcimgft  2674