Theorem r19.2mOLD 3502

Description: Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1631). The restricted version is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.) Obsolete version of r19.2m 3501 as of 7-Apr-2023. (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
r19.2mOLD	|- ( ( E. x x e. A /\ A. x e. A ph ) -> E. x e. A ph )

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem r19.2mOLD

Step	Hyp	Ref	Expression
1		df-ral 2453	. . . 4 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
2		exintr 1627	. . . 4 |- ( A. x ( x e. A -> ph ) -> ( E. x x e. A -> E. x ( x e. A /\ ph ) ) )
3	1, 2	sylbi 120	. . 3 |- ( A. x e. A ph -> ( E. x x e. A -> E. x ( x e. A /\ ph ) ) )
4		df-rex 2454	. . 3 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
5	3, 4	syl6ibr 161	. 2 |- ( A. x e. A ph -> ( E. x x e. A -> E. x e. A ph ) )
6	5	impcom 124	1 |- ( ( E. x x e. A /\ A. x e. A ph ) -> E. x e. A ph )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1346   E.wex 1485   e. wcel 2141   A.wral 2448   E.wrex 2449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-ral 2453  df-rex 2454

This theorem is referenced by: (None)