ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  r19.29vva Unicode version

Theorem r19.29vva 2473
Description: A commonly used pattern based on r19.29 2467, version with two restricted quantifiers. (Contributed by Thierry Arnoux, 26-Nov-2017.)
Hypotheses
Ref Expression
r19.29vva.1  |-  ( ( ( ( ph  /\  x  e.  A )  /\  y  e.  B
)  /\  ps )  ->  ch )
r19.29vva.2  |-  ( ph  ->  E. x  e.  A  E. y  e.  B  ps )
Assertion
Ref Expression
r19.29vva  |-  ( ph  ->  ch )
Distinct variable groups:    y, A    x, y, ch    ph, x, y
Allowed substitution hints:    ps( x, y)    A( x)    B( x, y)

Proof of Theorem r19.29vva
StepHypRef Expression
1 r19.29vva.1 . . . . . 6  |-  ( ( ( ( ph  /\  x  e.  A )  /\  y  e.  B
)  /\  ps )  ->  ch )
21ex 112 . . . . 5  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  B )  ->  ( ps  ->  ch ) )
32ralrimiva 2409 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  A. y  e.  B  ( ps  ->  ch ) )
43ralrimiva 2409 . . 3  |-  ( ph  ->  A. x  e.  A  A. y  e.  B  ( ps  ->  ch )
)
5 r19.29vva.2 . . 3  |-  ( ph  ->  E. x  e.  A  E. y  e.  B  ps )
64, 5r19.29d2r 2472 . 2  |-  ( ph  ->  E. x  e.  A  E. y  e.  B  ( ( ps  ->  ch )  /\  ps )
)
7 pm3.35 333 . . . . 5  |-  ( ( ps  /\  ( ps 
->  ch ) )  ->  ch )
87ancoms 259 . . . 4  |-  ( ( ( ps  ->  ch )  /\  ps )  ->  ch )
98rexlimivw 2446 . . 3  |-  ( E. y  e.  B  ( ( ps  ->  ch )  /\  ps )  ->  ch )
109rexlimivw 2446 . 2  |-  ( E. x  e.  A  E. y  e.  B  (
( ps  ->  ch )  /\  ps )  ->  ch )
116, 10syl 14 1  |-  ( ph  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    e. wcel 1409   A.wral 2323   E.wrex 2324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-ral 2328  df-rex 2329
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator