MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  s1eqd Unicode version

Theorem s1eqd 11717
Description: Equality theorem for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
Hypothesis
Ref Expression
s1eqd.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
s1eqd  |-  ( ph  ->  <" A ">  =  <" B "> )

Proof of Theorem s1eqd
StepHypRef Expression
1 s1eqd.1 . 2  |-  ( ph  ->  A  =  B )
2 s1eq 11716 . 2  |-  ( A  =  B  ->  <" A ">  =  <" B "> )
31, 2syl 16 1  |-  ( ph  ->  <" A ">  =  <" B "> )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649   <"cs1 11682
This theorem is referenced by:  swrds1  11750  s2eqd  11789  s3eqd  11790  s4eqd  11791  s5eqd  11792  s6eqd  11793  s7eqd  11794  s8eqd  11795  frmdgsum  14770  efgredlemc  15340  vrgpval  15362  vrgpinv  15364  frgpup2  15371  frgpup3lem  15372  psgnunilem5  27293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-iota 5385  df-fv 5429  df-s1 11688
  Copyright terms: Public domain W3C validator