Theorem r19.37 2646

Description: Restricted version of one direction of Theorem 19.37 of [Margaris] p. 90. In classical logic the converse would hold if A has at least one element, but that is not sufficient in intuitionistic logic. (Contributed by FL, 13-May-2012.) (Revised by Mario Carneiro, 11-Dec-2016.)

Hypothesis

Ref	Expression
r19.37.1	|- F/ x ph

Assertion

Ref	Expression
r19.37	|- ( E. x e. A ( ph -> ps ) -> ( ph -> E. x e. A ps ) )

Proof of Theorem r19.37

Step	Hyp	Ref	Expression
1		r19.37.1	. . 3 |- F/ x ph
2		ax-1 6	. . 3 |- ( ph -> ( x e. A -> ph ) )
3	1, 2	ralrimi 2565	. 2 |- ( ph -> A. x e. A ph )
4		r19.35-1 2644	. 2 |- ( E. x e. A ( ph -> ps ) -> ( A. x e. A ph -> E. x e. A ps ) )
5	3, 4	syl5 32	1 |- ( E. x e. A ( ph -> ps ) -> ( ph -> E. x e. A ps ) )

Syntax hints:   -> wi 4   F/wnf 1471   e. wcel 2164   A.wral 2472   E.wrex 2473

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-ral 2477  df-rex 2478

This theorem is referenced by:  r19.37av  2647