Theorem rspcda 3315

Description: Restricted specialization, using implicit substitution. (Contributed by Thierry Arnoux, 29-Jun-2020.)

Hypotheses

Ref	Expression
rspcdva.1	|- ( x = C -> ( ps <-> ch ) )
rspcdva.2	|- ( ph -> A. x e. A ps )
rspcdva.3	|- ( ph -> C e. A )
rspcda.1	|- F/ x ph

Assertion

Ref	Expression
rspcda	|- ( ph -> ch )

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

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

Proof of Theorem rspcda

Step	Hyp	Ref	Expression
1		rspcdva.3	. 2 |- ( ph -> C e. A )
2		rspcdva.2	. 2 |- ( ph -> A. x e. A ps )
3		rspcdva.1	. . 3 |- ( x = C -> ( ps <-> ch ) )
4	3	rspcv 3305	. 2 |- ( C e. A -> ( A. x e. A ps -> ch ) )
5	1, 2, 4	sylc 65	1 |- ( ph -> ch )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   F/wnf 1708   e. wcel 1990   A.wral 2912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202

This theorem is referenced by: (None)