Theorem r19.2mOLD 3539

Description: Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1652). 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 3538 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 2480	. . . 4 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
2		exintr 1648	. . . 4 |- ( A. x ( x e. A -> ph ) -> ( E. x x e. A -> E. x ( x e. A /\ ph ) ) )
3	1, 2	sylbi 121	. . 3 |- ( A. x e. A ph -> ( E. x x e. A -> E. x ( x e. A /\ ph ) ) )
4		df-rex 2481	. . 3 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
5	3, 4	imbitrrdi 162	. 2 |- ( A. x e. A ph -> ( E. x x e. A -> E. x e. A ph ) )
6	5	impcom 125	1 |- ( ( E. x x e. A /\ A. x e. A ph ) -> E. x e. A ph )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1362   E.wex 1506   e. wcel 2167   A.wral 2475   E.wrex 2476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-ral 2480  df-rex 2481

This theorem is referenced by: (None)