Theorem r19.2m 3495

Description: Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1626). 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 2227	. . . 4 |- ( x = z -> ( x e. A <-> z e. A ) )
2	1	cbvexv 1906	. . 3 |- ( E. x x e. A <-> E. z z e. A )
3		eleq1w 2227	. . . 4 |- ( z = y -> ( z e. A <-> y e. A ) )
4	3	cbvexv 1906	. . 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 2449	. . . . 5 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
7		exintr 1622	. . . . 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 2450	. . . 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 1341   E.wex 1480   e. wcel 2136   A.wral 2444   E.wrex 2445

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-clel 2161  df-ral 2449  df-rex 2450

This theorem is referenced by:  intssunim  3846  riinm  3938  iinexgm  4133  xpiindim  4741  cnviinm  5145  eusvobj2  5828  iinerm  6573  suplocexprlemml  7657  rexfiuz  10931  r19.2uz  10935  climuni  11234  pc2dvds  12261  cncnp2m  12871