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Theorem ideq2 36581
Description: For sets, the identity binary relation is the same as equality. (Contributed by Peter Mazsa, 24-Jun-2020.) (Revised by Peter Mazsa, 18-Dec-2021.)
Assertion
Ref Expression
ideq2 (𝐴𝑉 → (𝐴 I 𝐵𝐴 = 𝐵))

Proof of Theorem ideq2
StepHypRef Expression
1 brid 36580 . 2 (𝐴 I 𝐵𝐵 I 𝐴)
2 ideqg 5793 . . 3 (𝐴𝑉 → (𝐵 I 𝐴𝐵 = 𝐴))
3 eqcom 2743 . . 3 (𝐵 = 𝐴𝐴 = 𝐵)
42, 3bitrdi 286 . 2 (𝐴𝑉 → (𝐵 I 𝐴𝐴 = 𝐵))
51, 4bitrid 282 1 (𝐴𝑉 → (𝐴 I 𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1540  wcel 2105   class class class wbr 5092   I cid 5517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-br 5093  df-opab 5155  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628
This theorem is referenced by:  br1cossinidres  36724  br1cossxrnidres  36726
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