Theorem r19.2m 3454

Description: Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1618). The restricted version is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.) (Revised by Jim Kingdon, 7-Apr-2023.)

Assertion

Ref	Expression
r19.2m	|- ( ( E. y y e. A /\ A. x e. A ph ) -> E. x e. A ph )

Distinct variable groups:   x, A   y, A

Allowed substitution hints:   ph( x, y)

Proof of Theorem r19.2m

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1w 2201	. . . 4 |- ( x = z -> ( x e. A <-> z e. A ) )
2	1	cbvexv 1891	. . 3 |- ( E. x x e. A <-> E. z z e. A )
3		eleq1w 2201	. . . 4 |- ( z = y -> ( z e. A <-> y e. A ) )
4	3	cbvexv 1891	. . 3 |- ( E. z z e. A <-> E. y y e. A )
5	2, 4	bitri 183	. 2 |- ( E. x x e. A <-> E. y y e. A )
6		df-ral 2422	. . . . 5 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
7		exintr 1614	. . . . 5 |- ( A. x ( x e. A -> ph ) -> ( E. x x e. A -> E. x ( x e. A /\ ph ) ) )
8	6, 7	sylbi 120	. . . 4 |- ( A. x e. A ph -> ( E. x x e. A -> E. x ( x e. A /\ ph ) ) )
9		df-rex 2423	. . . 4 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
10	8, 9	syl6ibr 161	. . 3 |- ( A. x e. A ph -> ( E. x x e. A -> E. x e. A ph ) )
11	10	impcom 124	. 2 |- ( ( E. x x e. A /\ A. x e. A ph ) -> E. x e. A ph )
12	5, 11	sylanbr 283	1 |- ( ( E. y y e. A /\ A. x e. A ph ) -> E. x e. A ph )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1330   E.wex 1469   e. wcel 1481   A.wral 2417   E.wrex 2418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515

This theorem depends on definitions:  df-bi 116  df-clel 2136  df-ral 2422  df-rex 2423

This theorem is referenced by:  intssunim  3801  riinm  3893  iinexgm  4087  xpiindim  4684  cnviinm  5088  eusvobj2  5768  iinerm  6509  suplocexprlemml  7548  rexfiuz  10793  r19.2uz  10797  climuni  11094  cncnp2m  12439