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Theorem sloteq 11978
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 5421 . . 3 (𝐴 = 𝐵 → (𝑓𝐴) = (𝑓𝐵))
21mpteq2dv 4019 . 2 (𝐴 = 𝐵 → (𝑓 ∈ V ↦ (𝑓𝐴)) = (𝑓 ∈ V ↦ (𝑓𝐵)))
3 df-slot 11977 . 2 Slot 𝐴 = (𝑓 ∈ V ↦ (𝑓𝐴))
4 df-slot 11977 . 2 Slot 𝐵 = (𝑓 ∈ V ↦ (𝑓𝐵))
52, 3, 43eqtr4g 2197 1 (𝐴 = 𝐵 → Slot 𝐴 = Slot 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1331  Vcvv 2686  cmpt 3989  cfv 5123  Slot cslot 11972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-iota 5088  df-fv 5131  df-slot 11977
This theorem is referenced by:  ndxid  11997
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