MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  spimeh Unicode version

Theorem spimeh 1734
Description: Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.)
Hypotheses
Ref Expression
spimeh.1  |-  ( ph  ->  A. x ph )
spimeh.2  |-  ( x  =  z  ->  ( ph  ->  ps ) )
Assertion
Ref Expression
spimeh  |-  ( ph  ->  E. x ps )
Distinct variable group:    x, z
Allowed substitution hints:    ph( x, z)    ps( x, z)

Proof of Theorem spimeh
StepHypRef Expression
1 ax9v 1645 . . . 4  |-  -.  A. x  -.  x  =  z
2 id 19 . . . . . . 7  |-  ( ph  ->  ph )
32hbth 1542 . . . . . . . 8  |-  ( (
ph  ->  ph )  ->  A. x
( ph  ->  ph )
)
4 hba1 1731 . . . . . . . . 9  |-  ( A. x  -.  ps  ->  A. x A. x  -.  ps )
54a1i 10 . . . . . . . 8  |-  ( (
ph  ->  ph )  ->  ( A. x  -.  ps  ->  A. x A. x  -.  ps ) )
6 spimeh.1 . . . . . . . . . 10  |-  ( ph  ->  A. x ph )
76hbn 1732 . . . . . . . . 9  |-  ( -. 
ph  ->  A. x  -.  ph )
87a1i 10 . . . . . . . 8  |-  ( (
ph  ->  ph )  ->  ( -.  ph  ->  A. x  -.  ph ) )
93, 5, 8hbimd 1733 . . . . . . 7  |-  ( (
ph  ->  ph )  ->  (
( A. x  -.  ps  ->  -.  ph )  ->  A. x ( A. x  -.  ps  ->  -.  ph )
) )
102, 9ax-mp 8 . . . . . 6  |-  ( ( A. x  -.  ps  ->  -.  ph )  ->  A. x ( A. x  -.  ps  ->  -.  ph )
)
1110hbn 1732 . . . . 5  |-  ( -.  ( A. x  -.  ps  ->  -.  ph )  ->  A. x  -.  ( A. x  -.  ps  ->  -. 
ph ) )
12 spimeh.2 . . . . . . 7  |-  ( x  =  z  ->  ( ph  ->  ps ) )
13 sp 1728 . . . . . . 7  |-  ( A. x  -.  ps  ->  -.  ps )
1412, 13nsyli 133 . . . . . 6  |-  ( x  =  z  ->  ( A. x  -.  ps  ->  -. 
ph ) )
1514con3i 127 . . . . 5  |-  ( -.  ( A. x  -.  ps  ->  -.  ph )  ->  -.  x  =  z
)
1611, 15alrimih 1555 . . . 4  |-  ( -.  ( A. x  -.  ps  ->  -.  ph )  ->  A. x  -.  x  =  z )
171, 16mt3 171 . . 3  |-  ( A. x  -.  ps  ->  -.  ph )
1817con2i 112 . 2  |-  ( ph  ->  -.  A. x  -.  ps )
19 df-ex 1532 . 2  |-  ( E. x ps  <->  -.  A. x  -.  ps )
2018, 19sylibr 203 1  |-  ( ph  ->  E. x ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1530   E.wex 1531
This theorem is referenced by:  ax12olem1  1880  ax10lem2  1890
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-ex 1532
  Copyright terms: Public domain W3C validator