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Theorem bj-evaleq 34257
Description: Equality theorem for the Slot construction. This is currently a duplicate of sloteq 16476 but may diverge from it if/when a token Eval is introduced for evaluation in order to separate it from Slot and any of its possible modifications. (Contributed by BJ, 27-Dec-2021.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-evaleq (𝐴 = 𝐵 → Slot 𝐴 = Slot 𝐵)

Proof of Theorem bj-evaleq
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6663 . . 3 (𝐴 = 𝐵 → (𝑓𝐴) = (𝑓𝐵))
21mpteq2dv 5153 . 2 (𝐴 = 𝐵 → (𝑓 ∈ V ↦ (𝑓𝐴)) = (𝑓 ∈ V ↦ (𝑓𝐵)))
3 df-slot 16475 . 2 Slot 𝐴 = (𝑓 ∈ V ↦ (𝑓𝐴))
4 df-slot 16475 . 2 Slot 𝐵 = (𝑓 ∈ V ↦ (𝑓𝐵))
52, 3, 43eqtr4g 2878 1 (𝐴 = 𝐵 → Slot 𝐴 = Slot 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  Vcvv 3492  cmpt 5137  cfv 6348  Slot cslot 16470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-iota 6307  df-fv 6356  df-slot 16475
This theorem is referenced by: (None)
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