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

Theorem r19.29an 2612
Description: A commonly used pattern based on r19.29 2607. (Contributed by Thierry Arnoux, 29-Dec-2019.)
Hypothesis
Ref Expression
r19.29an.1  |-  ( ( ( ph  /\  x  e.  A )  /\  ps )  ->  ch )
Assertion
Ref Expression
r19.29an  |-  ( (
ph  /\  E. x  e.  A  ps )  ->  ch )
Distinct variable groups:    ch, x    ph, x
Allowed substitution hints:    ps( x)    A( x)

Proof of Theorem r19.29an
StepHypRef Expression
1 nfv 1521 . . 3  |-  F/ x ph
2 nfre1 2513 . . 3  |-  F/ x E. x  e.  A  ps
31, 2nfan 1558 . 2  |-  F/ x
( ph  /\  E. x  e.  A  ps )
4 r19.29an.1 . . 3  |-  ( ( ( ph  /\  x  e.  A )  /\  ps )  ->  ch )
54adantllr 478 . 2  |-  ( ( ( ( ph  /\  E. x  e.  A  ps )  /\  x  e.  A
)  /\  ps )  ->  ch )
6 simpr 109 . 2  |-  ( (
ph  /\  E. x  e.  A  ps )  ->  E. x  e.  A  ps )
73, 5, 6r19.29af 2611 1  |-  ( (
ph  /\  E. x  e.  A  ps )  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 2141   E.wrex 2449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527  ax-i5r 1528
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-ral 2453  df-rex 2454
This theorem is referenced by:  exmidontriimlem2  7199  summodclem2  11345
  Copyright terms: Public domain W3C validator