Theorem bj-evaleq 37741

Description: Equality theorem for the Slot construction. This is currently a duplicate of sloteq 17249 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.

Step	Hyp	Ref	Expression
1		fveq2 6881	. . 3 ⊢ (𝐴 = 𝐵 → (𝑓‘𝐴) = (𝑓‘𝐵))
2	1	mpteq2dv 5204	. 2 ⊢ (𝐴 = 𝐵 → (𝑓 ∈ V ↦ (𝑓‘𝐴)) = (𝑓 ∈ V ↦ (𝑓‘𝐵)))
3		df-slot 17248	. 2 ⊢ Slot 𝐴 = (𝑓 ∈ V ↦ (𝑓‘𝐴))
4		df-slot 17248	. 2 ⊢ Slot 𝐵 = (𝑓 ∈ V ↦ (𝑓‘𝐵))
5	2, 3, 4	3eqtr4g 2822	1 ⊢ (𝐴 = 𝐵 → Slot 𝐴 = Slot 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1569  Vcvv 3454   ↦ cmpt 5191  ‘cfv 6536  Slot cslot 17247

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-iota 6492  df-fv 6544  df-slot 17248

This theorem is used by: (None)