Theorem rspcdf 42424

Description: Restricted specialization, using implicit substitution. (Contributed by Emmett Weisz, 16-Jan-2020.)

Hypotheses

Ref	Expression
rspcdf.1	|- F/ x ph
rspcdf.2	|- F/ x ch
rspcdf.3	|- ( ph -> A e. B )
rspcdf.4	|- ( ( ph /\ x = A ) -> ( ps <-> ch ) )

Assertion

Ref	Expression
rspcdf	|- ( ph -> ( A. x e. B ps -> ch ) )

Distinct variable groups:   x, A   x, B

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

Proof of Theorem rspcdf

Step	Hyp	Ref	Expression
1		rspcdf.1	. . 3 |- F/ x ph
2		rspcdf.4	. . . 4 |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )
3	2	ex 450	. . 3 |- ( ph -> ( x = A -> ( ps <-> ch ) ) )
4	1, 3	alrimi 2082	. 2 |- ( ph -> A. x ( x = A -> ( ps <-> ch ) ) )
5		rspcdf.3	. 2 |- ( ph -> A e. B )
6		rspcdf.2	. . 3 |- F/ x ch
7	6	rspct 3302	. 2 |- ( A. x ( x = A -> ( ps <-> ch ) ) -> ( A e. B -> ( A. x e. B ps -> ch ) ) )
8	4, 5, 7	sylc 65	1 |- ( ph -> ( A. x e. B ps -> ch ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = 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)