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

Theorem equsal 1900
Description: A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 3-Oct-2016.)
Hypotheses
Ref Expression
equsal.1  |-  F/ x ps
equsal.2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
equsal  |-  ( A. x ( x  =  y  ->  ph )  <->  ps )

Proof of Theorem equsal
StepHypRef Expression
1 equsal.2 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
2 equsal.1 . . . . . 6  |-  F/ x ps
3219.3 1781 . . . . 5  |-  ( A. x ps  <->  ps )
41, 3syl6bbr 254 . . . 4  |-  ( x  =  y  ->  ( ph 
<-> 
A. x ps )
)
54pm5.74i 236 . . 3  |-  ( ( x  =  y  ->  ph )  <->  ( x  =  y  ->  A. x ps ) )
65albii 1553 . 2  |-  ( A. x ( x  =  y  ->  ph )  <->  A. x
( x  =  y  ->  A. x ps )
)
72nfri 1742 . . . . 5  |-  ( ps 
->  A. x ps )
87a1d 22 . . . 4  |-  ( ps 
->  ( x  =  y  ->  A. x ps )
)
92, 8alrimi 1745 . . 3  |-  ( ps 
->  A. x ( x  =  y  ->  A. x ps ) )
10 ax9o 1890 . . 3  |-  ( A. x ( x  =  y  ->  A. x ps )  ->  ps )
119, 10impbii 180 . 2  |-  ( ps  <->  A. x ( x  =  y  ->  A. x ps ) )
126, 11bitr4i 243 1  |-  ( A. x ( x  =  y  ->  ph )  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1527   F/wnf 1531
This theorem is referenced by:  equsalh  1901  equsex  1902  dvelimf  1937  sb6x  1969  asymref2  5060  intirr  5061  fun11  5315  pm13.192  27610
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-nf 1532
  Copyright terms: Public domain W3C validator