Theorem r19.36av 2657

Description: One direction of a restricted quantifier version of Theorem 19.36 of [Margaris] p. 90. In classical logic, the converse would hold if A has at least one element, but in intuitionistic logic, that is not a sufficient condition. (Contributed by NM, 22-Oct-2003.)

Assertion

Ref	Expression
r19.36av	|- ( E. x e. A ( ph -> ps ) -> ( A. x e. A ph -> ps ) )

Distinct variable group:   ps, x

Allowed substitution hints:   ph( x)   A( x)

Proof of Theorem r19.36av

Step	Hyp	Ref	Expression
1		r19.35-1 2656	. 2 |- ( E. x e. A ( ph -> ps ) -> ( A. x e. A ph -> E. x e. A ps ) )
2		idd 21	. . . 4 |- ( x e. A -> ( ps -> ps ) )
3	2	rexlimiv 2617	. . 3 |- ( E. x e. A ps -> ps )
4	3	imim2i 12	. 2 |- ( ( A. x e. A ph -> E. x e. A ps ) -> ( A. x e. A ph -> ps ) )
5	1, 4	syl 14	1 |- ( E. x e. A ( ph -> ps ) -> ( A. x e. A ph -> ps ) )

Syntax hints:   -> wi 4   e. wcel 2176   A.wral 2484   E.wrex 2485

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 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533  ax-17 1549  ax-ial 1557  ax-i5r 1558

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-ral 2489  df-rex 2490

This theorem is referenced by:  iinss  3979