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Theorem ssralv2 28294
Description: Quantification restricted to a subclass for two quantifiers. ssralv 3237 for two quantifiers. The proof of ssralv2 28294 was automatically generated by minimizing the automatically translated proof of ssralv2VD 28642. The automatic translation is by the tools program translatewithout_overwriting.cmd (Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ssralv2  |-  ( ( A  C_  B  /\  C  C_  D )  -> 
( A. x  e.  B  A. y  e.  D  ph  ->  A. x  e.  A  A. y  e.  C  ph ) )
Distinct variable groups:    x, A    x, B    x, C    y, C    x, D    y, D
Allowed substitution hints:    ph( x, y)    A( y)    B( y)

Proof of Theorem ssralv2
StepHypRef Expression
1 nfv 1605 . 2  |-  F/ x
( A  C_  B  /\  C  C_  D )
2 nfra1 2593 . 2  |-  F/ x A. x  e.  B  A. y  e.  D  ph
3 ssralv 3237 . . . . . 6  |-  ( A 
C_  B  ->  ( A. x  e.  B  A. y  e.  D  ph 
->  A. x  e.  A  A. y  e.  D  ph ) )
43adantr 451 . . . . 5  |-  ( ( A  C_  B  /\  C  C_  D )  -> 
( A. x  e.  B  A. y  e.  D  ph  ->  A. x  e.  A  A. y  e.  D  ph ) )
5 df-ral 2548 . . . . 5  |-  ( A. x  e.  A  A. y  e.  D  ph  <->  A. x
( x  e.  A  ->  A. y  e.  D  ph ) )
64, 5syl6ib 217 . . . 4  |-  ( ( A  C_  B  /\  C  C_  D )  -> 
( A. x  e.  B  A. y  e.  D  ph  ->  A. x
( x  e.  A  ->  A. y  e.  D  ph ) ) )
7 sp 1716 . . . 4  |-  ( A. x ( x  e.  A  ->  A. y  e.  D  ph )  -> 
( x  e.  A  ->  A. y  e.  D  ph ) )
86, 7syl6 29 . . 3  |-  ( ( A  C_  B  /\  C  C_  D )  -> 
( A. x  e.  B  A. y  e.  D  ph  ->  (
x  e.  A  ->  A. y  e.  D  ph ) ) )
9 ssralv 3237 . . . 4  |-  ( C 
C_  D  ->  ( A. y  e.  D  ph 
->  A. y  e.  C  ph ) )
109adantl 452 . . 3  |-  ( ( A  C_  B  /\  C  C_  D )  -> 
( A. y  e.  D  ph  ->  A. y  e.  C  ph ) )
118, 10syl6d 64 . 2  |-  ( ( A  C_  B  /\  C  C_  D )  -> 
( A. x  e.  B  A. y  e.  D  ph  ->  (
x  e.  A  ->  A. y  e.  C  ph ) ) )
121, 2, 11ralrimd 2631 1  |-  ( ( A  C_  B  /\  C  C_  D )  -> 
( A. x  e.  B  A. y  e.  D  ph  ->  A. x  e.  A  A. y  e.  C  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1527    e. wcel 1684   A.wral 2543    C_ wss 3152
This theorem is referenced by:  ordelordALT  28301  ordelordALTVD  28643
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-ral 2548  df-in 3159  df-ss 3166
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