Theorem spcedv 2862

Description: Existential specialization, using implicit substitution, deduction version. (Contributed by RP, 12-Aug-2020.)

Hypotheses

Ref	Expression
spcedv.1	|- ( ph -> X e. _V )
spcedv.2	|- ( ph -> ch )
spcedv.3	|- ( x = X -> ( ps <-> ch ) )

Assertion

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

Distinct variable groups:   x, X   ch, x

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

Proof of Theorem spcedv

Step	Hyp	Ref	Expression
1		spcedv.1	. 2 |- ( ph -> X e. _V )
2		spcedv.2	. 2 |- ( ph -> ch )
3		spcedv.3	. . 3 |- ( x = X -> ( ps <-> ch ) )
4	3	spcegv 2861	. 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 1373   E.wex 1515   e. wcel 2176   _Vcvv 2772

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774

This theorem is referenced by:  fprodseq  11894  gsumval2  13229