MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax6w Unicode version

Theorem ax6w 1703
Description: Weak version of ax-6 1715 from which we can prove any ax-6 1715 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)
Hypothesis
Ref Expression
ax6w.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
ax6w  |-  ( -. 
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 ax6w
StepHypRef Expression
1 ax6w.1 . 2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
21hbn1w 1692 1  |-  ( -. 
A. x ph  ->  A. x  -.  A. x ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176   A.wal 1530
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
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532
  Copyright terms: Public domain W3C validator