Theorem r19.2m 3309

Description: Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1529). 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 2311	. . . 4 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
2		exintr 1525	. . . 4 |- ( A. x ( x e. A -> ph ) -> ( E. x x e. A -> E. x ( x e. A /\ ph ) ) )
3	1, 2	sylbi 114	. . 3 |- ( A. x e. A ph -> ( E. x x e. A -> E. x ( x e. A /\ ph ) ) )
4		df-rex 2312	. . 3 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
5	3, 4	syl6ibr 151	. 2 |- ( A. x e. A ph -> ( E. x x e. A -> E. x e. A ph ) )
6	5	impcom 116	1 |- ( ( E. x x e. A /\ A. x e. A ph ) -> E. x e. A ph )

Syntax hints:   -> wi 4   /\ wa 97   A.wal 1241   E.wex 1381   e. wcel 1393   A.wral 2306   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-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427

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

This theorem is referenced by:  intssunim  3637  riinm  3729  trintssm  3870  iinexgm  3908  xpiindim  4473  cnviinm  4859  eusvobj2  5498  iinerm  6178  r19.2uz  9591  climuni  9814