Theorem spcedv 2819

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 2818	. 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 1348   E.wex 1485   e. wcel 2141   _Vcvv 2730

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732

This theorem is referenced by:  fprodseq  11546