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Theorem sloteq 17115
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 6891 . . 3 (𝐴 = 𝐵 → (𝑓𝐴) = (𝑓𝐵))
21mpteq2dv 5250 . 2 (𝐴 = 𝐵 → (𝑓 ∈ V ↦ (𝑓𝐴)) = (𝑓 ∈ V ↦ (𝑓𝐵)))
3 df-slot 17114 . 2 Slot 𝐴 = (𝑓 ∈ V ↦ (𝑓𝐴))
4 df-slot 17114 . 2 Slot 𝐵 = (𝑓 ∈ V ↦ (𝑓𝐵))
52, 3, 43eqtr4g 2797 1 (𝐴 = 𝐵 → Slot 𝐴 = Slot 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  Vcvv 3474  cmpt 5231  cfv 6543  Slot cslot 17113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-iota 6495  df-fv 6551  df-slot 17114
This theorem is referenced by:  ndxid  17129
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