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Theorem bj-evaleq 35251
Description: Equality theorem for the Slot construction. This is currently a duplicate of sloteq 16894 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 6766 . . 3 (𝐴 = 𝐵 → (𝑓𝐴) = (𝑓𝐵))
21mpteq2dv 5175 . 2 (𝐴 = 𝐵 → (𝑓 ∈ V ↦ (𝑓𝐴)) = (𝑓 ∈ V ↦ (𝑓𝐵)))
3 df-slot 16893 . 2 Slot 𝐴 = (𝑓 ∈ V ↦ (𝑓𝐴))
4 df-slot 16893 . 2 Slot 𝐵 = (𝑓 ∈ V ↦ (𝑓𝐵))
52, 3, 43eqtr4g 2803 1 (𝐴 = 𝐵 → Slot 𝐴 = Slot 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  Vcvv 3429  cmpt 5156  cfv 6426  Slot cslot 16892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3431  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5074  df-opab 5136  df-mpt 5157  df-iota 6384  df-fv 6434  df-slot 16893
This theorem is referenced by: (None)
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