Theorem 19.35-1 1515

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 1511	. . 3 |- ( ( A. x ph /\ E. x ( ph -> ps ) ) -> E. x ( ph /\ ( ph -> ps ) ) )
2		pm3.35 329	. . . 4 |- ( ( ph /\ ( ph -> ps ) ) -> ps )
3	2	eximi 1491	. . 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 109	1 |- ( E. x ( ph -> ps ) -> ( A. x ph -> E. x ps ) )

Syntax hints:   -> wi 4   /\ wa 97   A.wal 1241   E.wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  19.35i  1516  19.25  1517  19.36-1  1563  19.37-1  1564  spimt  1624  sbequi  1720