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Theorem rspc2edv 25066
Description: 2-variable restricted existential specialization, using implicit substitution. (rspc2ev 2905 with an antecedent.) (Contributed by FL, 2-Jul-2012.)
Hypotheses
Ref Expression
rspc2edv.1  |-  ( x  =  A  ->  ( ps 
<->  th ) )
rspc2edv.2  |-  ( y  =  B  ->  ( th 
<->  ch ) )
Assertion
Ref Expression
rspc2edv  |-  ( (
ph  /\  A  e.  C  /\  B  e.  D
)  ->  ( ch  ->  E. x  e.  C  E. y  e.  D  ps ) )
Distinct variable groups:    x, A, y    y, B    x, C    x, D, y    ch, y    ph, x, y    th, x
Allowed substitution hints:    ps( x, y)    ch( x)    th( y)    B( x)    C( y)

Proof of Theorem rspc2edv
StepHypRef Expression
1 rspc2edv.1 . . . . . . . 8  |-  ( x  =  A  ->  ( ps 
<->  th ) )
21anbi2d 684 . . . . . . 7  |-  ( x  =  A  ->  (
( ph  /\  ps )  <->  (
ph  /\  th )
) )
3 rspc2edv.2 . . . . . . . 8  |-  ( y  =  B  ->  ( th 
<->  ch ) )
43anbi2d 684 . . . . . . 7  |-  ( y  =  B  ->  (
( ph  /\  th )  <->  (
ph  /\  ch )
) )
52, 4rspc2ev 2905 . . . . . 6  |-  ( ( A  e.  C  /\  B  e.  D  /\  ( ph  /\  ch )
)  ->  E. x  e.  C  E. y  e.  D  ( ph  /\ 
ps ) )
6 simpr 447 . . . . . . . 8  |-  ( (
ph  /\  ps )  ->  ps )
76reximi 2663 . . . . . . 7  |-  ( E. y  e.  D  (
ph  /\  ps )  ->  E. y  e.  D  ps )
87reximi 2663 . . . . . 6  |-  ( E. x  e.  C  E. y  e.  D  ( ph  /\  ps )  ->  E. x  e.  C  E. y  e.  D  ps )
95, 8syl 15 . . . . 5  |-  ( ( A  e.  C  /\  B  e.  D  /\  ( ph  /\  ch )
)  ->  E. x  e.  C  E. y  e.  D  ps )
1093expia 1153 . . . 4  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( ( ph  /\  ch )  ->  E. x  e.  C  E. y  e.  D  ps )
)
1110exp4b 590 . . 3  |-  ( A  e.  C  ->  ( B  e.  D  ->  (
ph  ->  ( ch  ->  E. x  e.  C  E. y  e.  D  ps ) ) ) )
1211com3r 73 . 2  |-  ( ph  ->  ( A  e.  C  ->  ( B  e.  D  ->  ( ch  ->  E. x  e.  C  E. y  e.  D  ps )
) ) )
13123imp 1145 1  |-  ( (
ph  /\  A  e.  C  /\  B  e.  D
)  ->  ( ch  ->  E. x  e.  C  E. y  e.  D  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   E.wrex 2557
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-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-v 2803
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