Theorem r19.44av 2469

Description: One direction of a restricted quantifier version of Theorem 19.44 of [Margaris] p. 90. The other direction doesn't hold when A is empty. (Contributed by NM, 2-Apr-2004.)

Assertion

Ref	Expression
r19.44av	|- ( E. x e. A ( ph \/ ps ) -> ( E. x e. A ph \/ ps ) )

Distinct variable group:   ps, x

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

Proof of Theorem r19.44av

Step	Hyp	Ref	Expression
1		r19.43 2468	. 2 |- ( E. x e. A ( ph \/ ps ) <-> ( E. x e. A ph \/ E. x e. A ps ) )
2		idd 21	. . . 4 |- ( x e. A -> ( ps -> ps ) )
3	2	rexlimiv 2427	. . 3 |- ( E. x e. A ps -> ps )
4	3	orim2i 678	. 2 |- ( ( E. x e. A ph \/ E. x e. A ps ) -> ( E. x e. A ph \/ ps ) )
5	1, 4	sylbi 114	1 |- ( E. x e. A ( ph \/ ps ) -> ( E. x e. A ph \/ ps ) )

Syntax hints:   -> wi 4   \/ wo 629   e. wcel 1393   E.wrex 2307

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-ral 2311  df-rex 2312

This theorem is referenced by: (None)