ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  equs4 Unicode version

Theorem equs4 1654
Description: Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 20-May-2014.)
Assertion
Ref Expression
equs4  |-  ( A. x ( x  =  y  ->  ph )  ->  E. x ( x  =  y  /\  ph )
)

Proof of Theorem equs4
StepHypRef Expression
1 a9e 1627 . . 3  |-  E. x  x  =  y
2 19.29 1552 . . 3  |-  ( ( A. x ( x  =  y  ->  ph )  /\  E. x  x  =  y )  ->  E. x
( ( x  =  y  ->  ph )  /\  x  =  y )
)
31, 2mpan2 416 . 2  |-  ( A. x ( x  =  y  ->  ph )  ->  E. x ( ( x  =  y  ->  ph )  /\  x  =  y
) )
4 ancl 311 . . . 4  |-  ( ( x  =  y  ->  ph )  ->  ( x  =  y  ->  (
x  =  y  /\  ph ) ) )
54imp 122 . . 3  |-  ( ( ( x  =  y  ->  ph )  /\  x  =  y )  -> 
( x  =  y  /\  ph ) )
65eximi 1532 . 2  |-  ( E. x ( ( x  =  y  ->  ph )  /\  x  =  y
)  ->  E. x
( x  =  y  /\  ph ) )
73, 6syl 14 1  |-  ( A. x ( x  =  y  ->  ph )  ->  E. x ( x  =  y  /\  ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   A.wal 1283    = wceq 1285   E.wex 1422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  sb2  1691  equs45f  1724
  Copyright terms: Public domain W3C validator