Theorem r19.45av 2514

Description: Restricted version of one direction of Theorem 19.45 of [Margaris] p. 90. (The other direction doesn't hold when A is empty.) (Contributed by NM, 2-Apr-2004.)

Assertion

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

Step	Hyp	Ref	Expression
1		r19.43 2512	. 2 |- ( E. x e. A ( ph \/ ps ) <-> ( E. x e. A ph \/ E. x e. A ps ) )
2		idd 21	. . . 4 |- ( x e. A -> ( ph -> ph ) )
3	2	rexlimiv 2471	. . 3 |- ( E. x e. A ph -> ph )
4	3	orim1i 709	. 2 |- ( ( E. x e. A ph \/ E. x e. A ps ) -> ( ph \/ E. x e. A ps ) )
5	1, 4	sylbi 119	1 |- ( E. x e. A ( ph \/ ps ) -> ( ph \/ E. x e. A ps ) )

Syntax hints:   -> wi 4   \/ wo 661   e. wcel 1433   E.wrex 2349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-ral 2353  df-rex 2354

This theorem is referenced by: (None)