Theorem r19.2m 3303

Description: Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1526). 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 2305	. . . 4 |- (A.x e. A ph <-> A.x(x e. A -> ph))
2		exintr 1522	. . . 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 2306	. . 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 1240  E.wex 1378   e. wcel 1390  A.wral 2300  E.wrex 2301

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 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-ial 1424

This theorem depends on definitions:  df-bi 110  df-ral 2305  df-rex 2306

This theorem is referenced by:  intssunim  3628  riinm  3720  trintssm  3861  iinexgm  3899  xpiindim  4416  cnviinm  4802  eusvobj2  5441  iinerm  6114