Theorem r19.37v 3087

Description: Restricted quantifier version of one direction of 19.37v 1910. (The other direction holds iff A is nonempty, see r19.37zv 4067.) (Contributed by NM, 2-Apr-2004.)

Assertion

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

Distinct variable group:   ph, x

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

Proof of Theorem r19.37v

Step	Hyp	Ref	Expression
1		nfv 1843	. 2 |- F/ x ph
2	1	r19.37 3086	1 |- ( E. x e. A ( ph -> ps ) -> ( ph -> E. x e. A ps ) )

Syntax hints:   -> wi 4   E.wrex 2913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-nf 1710  df-ral 2917  df-rex 2918

This theorem is referenced by:  ssiun  4562  isucn2  22083