Theorem spcedv 2827

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 2826	. 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 1353   E.wex 1492   e. wcel 2148   _Vcvv 2738

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740

This theorem is referenced by:  fprodseq  11591