Theorem elabd 2971

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 2913	. 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 1402   E.wex 1545   e. wcel 2209   _Vcvv 2821

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823

This theorem is referenced by:  uchoice  6361  elmpom  6464  en2prd  7096  dom1o  7106  ntrivcvgap0  12294  ssomct  13314  gsumvalfi  14129  wlkvtxiedg  16500  wlkvtxiedgg  16501  dceqnconst  17015  dcapnconst  17016