Theorem r19.2m 3614

Description: Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1691). 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 2299	. . . 4 |- ( x = z -> ( x e. A <-> z e. A ) )
2	1	cbvexv 1974	. . 3 |- ( E. x x e. A <-> E. z z e. A )
3		eleq1w 2299	. . . 4 |- ( z = y -> ( z e. A <-> y e. A ) )
4	3	cbvexv 1974	. . 3 |- ( E. z z e. A <-> E. y y e. A )
5	2, 4	bitri 184	. 2 |- ( E. x x e. A <-> E. y y e. A )
6		df-ral 2533	. . . . 5 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
7		exintr 1687	. . . . 5 |- ( A. x ( x e. A -> ph ) -> ( E. x x e. A -> E. x ( x e. A /\ ph ) ) )
8	6, 7	sylbi 121	. . . 4 |- ( A. x e. A ph -> ( E. x x e. A -> E. x ( x e. A /\ ph ) ) )
9		df-rex 2534	. . . 4 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
10	8, 9	imbitrrdi 162	. . 3 |- ( A. x e. A ph -> ( E. x x e. A -> E. x e. A ph ) )
11	10	impcom 125	. 2 |- ( ( E. x x e. A /\ A. x e. A ph ) -> E. x e. A ph )
12	5, 11	sylanbr 285	1 |- ( ( E. y y e. A /\ A. x e. A ph ) -> E. x e. A ph )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   A.wal 1400   E.wex 1545   e. wcel 2209   A.wral 2528   E.wrex 2529

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587

This proof depends on definitions:  df-bi 117  df-clel 2234  df-ral 2533  df-rex 2534

This theorem is used by:  intssunim  3992  riinm  4085  iinexgm  4290  xpiindim  4917  cnviinm  5329  eusvobj2  6071  iinerm  6881  suplocexprlemml  8083  rexfiuz  11755  r19.2uz  11759  climuni  12059  pc2dvds  13109  issubg4m  13996  cncnp2m  15332  alsralrex  17153