Theorem 19.35-1 1612

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 1608	. . 3 |- ( ( A. x ph /\ E. x ( ph -> ps ) ) -> E. x ( ph /\ ( ph -> ps ) ) )
2		pm3.35 345	. . . 4 |- ( ( ph /\ ( ph -> ps ) ) -> ps )
3	2	eximi 1588	. . 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 115	1 |- ( E. x ( ph -> ps ) -> ( A. x ph -> E. x ps ) )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1341   E.wex 1480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-ial 1522

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  19.35i  1613  19.25  1614  19.36-1  1661  19.37-1  1662  spimt  1724  sbequi  1827