Theorem spc2d 29313

Description: Specialization with 2 quantifiers, using implicit substitution. (Contributed by Thierry Arnoux, 23-Aug-2017.)

Hypotheses

Ref	Expression
spc2ed.x	|- F/ x ch
spc2ed.y	|- F/ y ch
spc2ed.1	|- ( ( ph /\ ( x = A /\ y = B ) ) -> ( ps <-> ch ) )

Assertion

Ref	Expression
spc2d	|- ( ( ph /\ ( A e. V /\ B e. W ) ) -> ( A. x A. y ps -> ch ) )

Distinct variable groups:   x, y, A   x, B, y   ph, x, y

Allowed substitution hints:   ps( x, y)   ch( x, y)   V( x, y)   W( x, y)

Proof of Theorem spc2d

Step	Hyp	Ref	Expression
1		2nalexn 1755	. . 3 |- ( -. A. x A. y ps <-> E. x E. y -. ps )
2	1	con1bii 346	. 2 |- ( -. E. x E. y -. ps <-> A. x A. y ps )
3		spc2ed.x	. . . . 5 |- F/ x ch
4	3	nfn 1784	. . . 4 |- F/ x -. ch
5		spc2ed.y	. . . . 5 |- F/ y ch
6	5	nfn 1784	. . . 4 |- F/ y -. ch
7		spc2ed.1	. . . . 5 |- ( ( ph /\ ( x = A /\ y = B ) ) -> ( ps <-> ch ) )
8	7	notbid 308	. . . 4 |- ( ( ph /\ ( x = A /\ y = B ) ) -> ( -. ps <-> -. ch ) )
9	4, 6, 8	spc2ed 29312	. . 3 |- ( ( ph /\ ( A e. V /\ B e. W ) ) -> ( -. ch -> E. x E. y -. ps ) )
10	9	con1d 139	. 2 |- ( ( ph /\ ( A e. V /\ B e. W ) ) -> ( -. E. x E. y -. ps -> ch ) )
11	2, 10	syl5bir 233	1 |- ( ( ph /\ ( A e. V /\ B e. W ) ) -> ( A. x A. y ps -> ch ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   F/wnf 1708   e. wcel 1990

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-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-v 3202

This theorem is referenced by: (None)