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Theorem sloteq 17152
Description: Equality theorem for the Slot construction. The converse holds if 𝐴 (or 𝐵) is a set. (Contributed by BJ, 27-Dec-2021.)
Assertion
Ref Expression
sloteq (𝐴 = 𝐵 → Slot 𝐴 = Slot 𝐵)

Proof of Theorem sloteq
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6897 . . 3 (𝐴 = 𝐵 → (𝑓𝐴) = (𝑓𝐵))
21mpteq2dv 5250 . 2 (𝐴 = 𝐵 → (𝑓 ∈ V ↦ (𝑓𝐴)) = (𝑓 ∈ V ↦ (𝑓𝐵)))
3 df-slot 17151 . 2 Slot 𝐴 = (𝑓 ∈ V ↦ (𝑓𝐴))
4 df-slot 17151 . 2 Slot 𝐵 = (𝑓 ∈ V ↦ (𝑓𝐵))
52, 3, 43eqtr4g 2793 1 (𝐴 = 𝐵 → Slot 𝐴 = Slot 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  Vcvv 3471  cmpt 5231  cfv 6548  Slot cslot 17150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-iota 6500  df-fv 6556  df-slot 17151
This theorem is referenced by:  ndxid  17166
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