Theorem r19.45v 3095

Description: Restricted quantifier version of one direction of 19.45 2107. The other direction holds when A is nonempty, see r19.45zv 4068. (Contributed by NM, 2-Apr-2004.)

Assertion

Ref	Expression
r19.45v	|- ( 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 r19.45v

Step	Hyp	Ref	Expression
1		r19.43 3093	. 2 |- ( E. x e. A ( ph \/ ps ) <-> ( E. x e. A ph \/ E. x e. A ps ) )
2		id 22	. . . 4 |- ( ph -> ph )
3	2	rexlimivw 3029	. . 3 |- ( E. x e. A ph -> ph )
4	3	orim1i 539	. 2 |- ( ( E. x e. A ph \/ E. x e. A ps ) -> ( ph \/ E. x e. A ps ) )
5	1, 4	sylbi 207	1 |- ( E. x e. A ( ph \/ ps ) -> ( ph \/ E. x e. A ps ) )

Syntax hints:   -> wi 4   \/ wo 383   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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-ral 2917  df-rex 2918

This theorem is referenced by: (None)