Theorem r19.21be 2933

Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 21-Nov-1994.)

Hypothesis

Ref	Expression
r19.21be.1	|- ( ph -> A. x e. A ps )

Assertion

Ref	Expression
r19.21be	|- A. x e. A ( ph -> ps )

Proof of Theorem r19.21be

Step	Hyp	Ref	Expression
1		r19.21be.1	. . . 4 |- ( ph -> A. x e. A ps )
2	1	r19.21bi 2932	. . 3 |- ( ( ph /\ x e. A ) -> ps )
3	2	expcom 451	. 2 |- ( x e. A -> ( ph -> ps ) )
4	3	rgen 2922	1 |- A. x e. A ( ph -> ps )

Syntax hints:   -> wi 4   e. wcel 1990   A.wral 2912

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-ral 2917

This theorem is referenced by:  bnj580  30983