Theorem hoeqi 28620

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 28619	. 2 |- ( ( S : ~H --> ~H /\ T : ~H --> ~H ) -> ( A. x e. ~H ( S ` x ) = ( T ` x ) <-> S = T ) )
4	1, 2, 3	mp2an 708	1 |- ( A. x e. ~H ( S ` x ) = ( T ` x ) <-> S = T )

Syntax hints:   <-> wb 196   = wceq 1483   A.wral 2912   -->wf 5884   ` cfv 5888   ~Hchil 27776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896

This theorem is referenced by:  hoaddcomi  28631  hodsi  28634  hoaddassi  28635  hocadddiri  28638  hocsubdiri  28639  hoaddid1i  28645  ho0coi  28647  hoid1i  28648  hoid1ri  28649  honegsubi  28655  hoddii  28848  pjsdii  29014  pjddii  29015  pjss1coi  29022  pjss2coi  29023  pjorthcoi  29028  pjscji  29029  pjtoi  29038  pjclem4  29058  pj3si  29066  pj3cor1i  29068