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Theorem ssralv2 28593
Description: Quantification restricted to a subclass for two quantifiers. ssralv 3250 for two quantifiers. The proof of ssralv2 28593 was automatically generated by minimizing the automatically translated proof of ssralv2VD 28958. 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 1609 . 2  |-  F/ x
( A  C_  B  /\  C  C_  D )
2 nfra1 2606 . 2  |-  F/ x A. x  e.  B  A. y  e.  D  ph
3 ssralv 3250 . . . . . 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 2561 . . . . 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 1728 . . . 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 3250 . . . 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 2644 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 1530    e. wcel 1696   A.wral 2556    C_ wss 3165
This theorem is referenced by:  ordelordALT  28600  ordelordALTVD  28959
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-ral 2561  df-in 3172  df-ss 3179
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