Theorem elabd 2829

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 2774	. 2 |- ( X e. _V -> ( ch -> E. x ps ) )
5	1, 2, 4	sylc 62	1 |- ( ph -> E. x ps )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1331   E.wex 1468   e. wcel 1480   _Vcvv 2686

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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688

This theorem is referenced by:  ntrivcvgap0  11323