Theorem elabd 2909

Description: Explicit demonstration the class { x | ps } is not empty by the example X. (Contributed by RP, 12-Aug-2020.)

Hypotheses

Ref	Expression
elab.xex	|- ( ph -> X e. _V )
elab.xmaj	|- ( ph -> ch )
elab.xsub	|- ( x = X -> ( ps <-> ch ) )

Assertion

Ref	Expression
elabd	|- ( ph -> E. x ps )

Distinct variable groups:   ch, x   x, X

Allowed substitution hints:   ph( x)   ps( x)

Proof of Theorem elabd

Step	Hyp	Ref	Expression
1		elab.xex	. 2 |- ( ph -> X e. _V )
2		elab.xmaj	. 2 |- ( ph -> ch )
3		elab.xsub	. . 3 |- ( x = X -> ( ps <-> ch ) )
4	3	spcegv 2852	. 2 |- ( X e. _V -> ( ch -> E. x ps ) )
5	1, 2, 4	sylc 62	1 |- ( ph -> E. x ps )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   E.wex 1506   e. wcel 2167   _Vcvv 2763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765

This theorem is referenced by:  uchoice  6195  ntrivcvgap0  11714  ssomct  12662  dceqnconst  15704  dcapnconst  15705