Theorem r19.2m 3329

Description: Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1569). The restricted version is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)

Assertion

Ref	Expression
r19.2m	|- ( ( 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 r19.2m

Step	Hyp	Ref	Expression
1		df-ral 2353	. . . 4 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
2		exintr 1565	. . . 4 |- ( A. x ( x e. A -> ph ) -> ( E. x x e. A -> E. x ( x e. A /\ ph ) ) )
3	1, 2	sylbi 119	. . 3 |- ( A. x e. A ph -> ( E. x x e. A -> E. x ( x e. A /\ ph ) ) )
4		df-rex 2354	. . 3 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
5	3, 4	syl6ibr 160	. 2 |- ( A. x e. A ph -> ( E. x x e. A -> E. x e. A ph ) )
6	5	impcom 123	1 |- ( ( E. x x e. A /\ A. x e. A ph ) -> E. x e. A ph )

Syntax hints:   -> wi 4   /\ wa 102   A.wal 1282   E.wex 1421   e. wcel 1433   A.wral 2348   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-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-ial 1467

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

This theorem is referenced by:  intssunim  3658  riinm  3750  trintssmOLD  3892  iinexgm  3929  xpiindim  4491  cnviinm  4879  eusvobj2  5518  iinerm  6201  rexfiuz  9875  r19.2uz  9879  climuni  10132