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

Theorem r19.29an 2599
Description: A commonly used pattern based on r19.29 2594. (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 1508 . . 3  |-  F/ x ph
2 nfre1 2500 . . 3  |-  F/ x E. x  e.  A  ps
31, 2nfan 1545 . 2  |-  F/ x
( ph  /\  E. x  e.  A  ps )
4 r19.29an.1 . . 3  |-  ( ( ( ph  /\  x  e.  A )  /\  ps )  ->  ch )
54adantllr 473 . 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 2598 1  |-  ( (
ph  /\  E. x  e.  A  ps )  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 2128   E.wrex 2436
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 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-17 1506  ax-ial 1514  ax-i5r 1515
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-ral 2440  df-rex 2441
This theorem is referenced by:  exmidontriimlem2  7159  summodclem2  11290
  Copyright terms: Public domain W3C validator