Theorem r19.2m 3373

Description: Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1575). 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 2365	. . . 4 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
2		exintr 1571	. . . 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 2366	. . 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 1288   E.wex 1427   e. wcel 1439   A.wral 2360   E.wrex 2361

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-4 1446  ax-ial 1473

This theorem depends on definitions:  df-bi 116  df-ral 2365  df-rex 2366

This theorem is referenced by:  intssunim  3716  riinm  3808  trintssmOLD  3959  iinexgm  3996  xpiindim  4586  cnviinm  4985  eusvobj2  5652  iinerm  6378  rexfiuz  10483  r19.2uz  10487  climuni  10742