Theorem spcedv 2895

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 2894	. 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 1397   E.wex 1540   e. wcel 2202   _Vcvv 2802

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804

This theorem is referenced by:  fprodseq  12145  gsumval2  13481