Theorem 19.35-1 1638

Description: Forward direction of Theorem 19.35 of [Margaris] p. 90. The converse holds for classical logic but not (for all propositions) in intuitionistic logic. (Contributed by Mario Carneiro, 2-Feb-2015.)

Assertion

Ref	Expression
19.35-1	|- ( E. x ( ph -> ps ) -> ( A. x ph -> E. x ps ) )

Proof of Theorem 19.35-1

Step	Hyp	Ref	Expression
1		19.29 1634	. . 3 |- ( ( A. x ph /\ E. x ( ph -> ps ) ) -> E. x ( ph /\ ( ph -> ps ) ) )
2		pm3.35 347	. . . 4 |- ( ( ph /\ ( ph -> ps ) ) -> ps )
3	2	eximi 1614	. . 3 |- ( E. x ( ph /\ ( ph -> ps ) ) -> E. x ps )
4	1, 3	syl 14	. 2 |- ( ( A. x ph /\ E. x ( ph -> ps ) ) -> E. x ps )
5	4	expcom 116	1 |- ( E. x ( ph -> ps ) -> ( A. x ph -> E. x ps ) )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1362   E.wex 1506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-ial 1548

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  19.35i  1639  19.25  1640  19.36-1  1687  19.37-1  1688  spimt  1750  sbequi  1853