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

Theorem alcomiw 1690
Description: Weak version of alcom 1723. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.)
Hypothesis
Ref Expression
alcomiw.1  |-  ( y  =  z  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
alcomiw  |-  ( A. x A. y ph  ->  A. y A. x ph )
Distinct variable groups:    y, z    x, y    ph, z    ps, y
Allowed substitution hints:    ph( x, y)    ps( x, z)

Proof of Theorem alcomiw
StepHypRef Expression
1 alcomiw.1 . . . . 5  |-  ( y  =  z  ->  ( ph 
<->  ps ) )
21biimpd 198 . . . 4  |-  ( y  =  z  ->  ( ph  ->  ps ) )
32cbvalivw 1660 . . 3  |-  ( A. y ph  ->  A. z ps )
43alimi 1549 . 2  |-  ( A. x A. y ph  ->  A. x A. z ps )
5 ax-17 1606 . 2  |-  ( A. x A. z ps  ->  A. y A. x A. z ps )
61biimprd 214 . . . . . 6  |-  ( y  =  z  ->  ( ps  ->  ph ) )
76equcoms 1666 . . . . 5  |-  ( z  =  y  ->  ( ps  ->  ph ) )
87spimvw 1657 . . . 4  |-  ( A. z ps  ->  ph )
98alimi 1549 . . 3  |-  ( A. x A. z ps  ->  A. x ph )
109alimi 1549 . 2  |-  ( A. y A. x A. z ps  ->  A. y A. x ph )
114, 5, 103syl 18 1  |-  ( A. x A. y ph  ->  A. y A. x ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1530
This theorem is referenced by:  hbalw  1695  ax7w  1704
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