MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hba1w Structured version   Unicode version

Theorem hba1w 1752
Description: Weak version of hba1 1829. See comments for ax10w 1763. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)
Hypothesis
Ref Expression
hbn1w.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
hba1w  |-  ( A. x ph  ->  A. x A. x ph )
Distinct variable groups:    ph, y    ps, x    x, y
Allowed substitution hints:    ph( x)    ps( y)

Proof of Theorem hba1w
StepHypRef Expression
1 hbn1w.1 . . . . . . 7  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
21cbvalvw 1747 . . . . . 6  |-  ( A. x ph  <->  A. y ps )
32a1i 11 . . . . 5  |-  ( x  =  y  ->  ( A. x ph  <->  A. y ps ) )
43notbid 294 . . . 4  |-  ( x  =  y  ->  ( -.  A. x ph  <->  -.  A. y ps ) )
54spw 1745 . . 3  |-  ( A. x  -.  A. x ph  ->  -.  A. x ph )
65con2i 120 . 2  |-  ( A. x ph  ->  -.  A. x  -.  A. x ph )
74hbn1w 1751 . 2  |-  ( -. 
A. x  -.  A. x ph  ->  A. x  -.  A. x  -.  A. x ph )
81hbn1w 1751 . . . 4  |-  ( -. 
A. x ph  ->  A. x  -.  A. x ph )
98con1i 129 . . 3  |-  ( -. 
A. x  -.  A. x ph  ->  A. x ph )
109alimi 1604 . 2  |-  ( A. x  -.  A. x  -.  A. x ph  ->  A. x A. x ph )
116, 7, 103syl 20 1  |-  ( A. x ph  ->  A. x A. x ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184   A.wal 1367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728
This theorem depends on definitions:  df-bi 185  df-ex 1587
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator