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Theorem hoeqi 23217
Description: Equality of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)
Hypotheses
Ref Expression
hoeq.1  |-  S : ~H
--> ~H
hoeq.2  |-  T : ~H
--> ~H
Assertion
Ref Expression
hoeqi  |-  ( A. x  e.  ~H  ( S `  x )  =  ( T `  x )  <->  S  =  T )
Distinct variable groups:    x, S    x, T

Proof of Theorem hoeqi
StepHypRef Expression
1 hoeq.1 . 2  |-  S : ~H
--> ~H
2 hoeq.2 . 2  |-  T : ~H
--> ~H
3 hoeq 23216 . 2  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( A. x  e. 
~H  ( S `  x )  =  ( T `  x )  <-> 
S  =  T ) )
41, 2, 3mp2an 654 1  |-  ( A. x  e.  ~H  ( S `  x )  =  ( T `  x )  <->  S  =  T )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1649   A.wral 2666   -->wf 5409   ` cfv 5413   ~Hchil 22375
This theorem is referenced by:  hoaddcomi  23228  hodsi  23231  hoaddassi  23232  hocadddiri  23235  hocsubdiri  23236  hoaddid1i  23242  ho0coi  23244  hoid1i  23245  hoid1ri  23246  honegsubi  23252  hoddii  23445  pjsdii  23611  pjddii  23612  pjss1coi  23619  pjss2coi  23620  pjorthcoi  23625  pjscji  23626  pjtoi  23635  pjclem4  23655  pj3si  23663  pj3cor1i  23665
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-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
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-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421
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