Theorem spcedv 2869

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 2868	. 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 1516   e. wcel 2178   _Vcvv 2776

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778

This theorem is referenced by:  fprodseq  12009  gsumval2  13344