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Theorem reximddv2 2902
Description: Double deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Thierry Arnoux, 15-Dec-2019.)
Hypotheses
Ref Expression
reximddv2.1  |-  ( ( ( ( ph  /\  x  e.  A )  /\  y  e.  B
)  /\  ps )  ->  ch )
reximddv2.2  |-  ( ph  ->  E. x  e.  A  E. y  e.  B  ps )
Assertion
Ref Expression
reximddv2  |-  ( ph  ->  E. x  e.  A  E. y  e.  B  ch )
Distinct variable groups:    y, A    ph, x, y
Allowed substitution hints:    ps( x, y)    ch( x, y)    A( x)    B( x, y)

Proof of Theorem reximddv2
StepHypRef Expression
1 reximddv2.1 . . . . 5  |-  ( ( ( ( ph  /\  x  e.  A )  /\  y  e.  B
)  /\  ps )  ->  ch )
21ex 435 . . . 4  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  B )  ->  ( ps  ->  ch ) )
32reximdva 2900 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( E. y  e.  B  ps  ->  E. y  e.  B  ch ) )
43impr 623 . 2  |-  ( (
ph  /\  ( x  e.  A  /\  E. y  e.  B  ps )
)  ->  E. y  e.  B  ch )
5 reximddv2.2 . 2  |-  ( ph  ->  E. x  e.  A  E. y  e.  B  ps )
64, 5reximddv 2901 1  |-  ( ph  ->  E. x  e.  A  E. y  e.  B  ch )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    e. wcel 1868   E.wrex 2776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1660  df-ral 2780  df-rex 2781
This theorem is referenced by:  prmgaplem8  15016  cpmadugsumfi  19888  cpmidg2sum  19891  cayhamlem4  19899  ltgseg  24628  cgraswap  24849  cgracom  24851  cgratr  24852  dfcgra2  24858  xrofsup  28347  prmunb2  36517
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