Theorem hoeqi 22341

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

Step	Hyp	Ref	Expression
1		hoeq.1	. 2 |- S : ~H --> ~H
2		hoeq.2	. 2 |- T : ~H --> ~H
3		hoeq 22340	. 2 |- ( ( S : ~H --> ~H /\ T : ~H --> ~H ) -> ( A. x e. ~H ( S ` x ) = ( T ` x ) <-> S = T ) )
4	1, 2, 3	mp2an 653	1 |- ( A. x e. ~H ( S ` x ) = ( T ` x ) <-> S = T )

Syntax hints:   <-> wb 176   = wceq 1623   A.wral 2543   -->wf 5251   ` cfv 5255   ~Hchil 21499

This theorem is referenced by:  hoaddcomi  22352  hodsi  22355  hoaddassi  22356  hocadddiri  22359  hocsubdiri  22360  hoaddid1i  22366  ho0coi  22368  hoid1i  22369  hoid1ri  22370  honegsubi  22376  hoddii  22569  pjsdii  22735  pjddii  22736  pjss1coi  22743  pjss2coi  22744  pjorthcoi  22749  pjscji  22750  pjtoi  22759  pjclem4  22779  pj3si  22787  pj3cor1i  22789

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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263