Theorem spcdv 2865

Description: Rule of specialization, using implicit substitution. Analogous to rspcdv 2887. (Contributed by David Moews, 1-May-2017.)

Hypotheses

Ref	Expression
spcimdv.1	|- ( ph -> A e. B )
spcdv.2	|- ( ( ph /\ x = A ) -> ( ps <-> ch ) )

Assertion

Ref	Expression
spcdv	|- ( ph -> ( A. x ps -> ch ) )

Distinct variable groups:   x, A   ph, x   ch, x

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

Proof of Theorem spcdv

Step	Hyp	Ref	Expression
1		spcimdv.1	. 2 |- ( ph -> A e. B )
2		spcdv.2	. . 3 |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )
3	2	biimpd 144	. 2 |- ( ( ph /\ x = A ) -> ( ps -> ch ) )
4	1, 3	spcimdv 2864	1 |- ( ph -> ( A. x ps -> ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1371   = wceq 1373   e. wcel 2178

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: (None)