Theorem bj-evaleq 37426

Description: Equality theorem for the Slot construction. This is currently a duplicate of sloteq 17147 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 6830	. . 3 ⊢ (𝐴 = 𝐵 → (𝑓‘𝐴) = (𝑓‘𝐵))
2	1	mpteq2dv 5169	. 2 ⊢ (𝐴 = 𝐵 → (𝑓 ∈ V ↦ (𝑓‘𝐴)) = (𝑓 ∈ V ↦ (𝑓‘𝐵)))
3		df-slot 17146	. 2 ⊢ Slot 𝐴 = (𝑓 ∈ V ↦ (𝑓‘𝐴))
4		df-slot 17146	. 2 ⊢ Slot 𝐵 = (𝑓 ∈ V ↦ (𝑓‘𝐵))
5	2, 3, 4	3eqtr4g 2796	1 ⊢ (𝐴 = 𝐵 → Slot 𝐴 = Slot 𝐵)

Syntax hints:   → wi 4   = wceq 1543  Vcvv 3428   ↦ cmpt 5156  ‘cfv 6488  Slot cslot 17145

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 1970  ax-7 2011  ax-8 2117  ax-9 2125  ax-ext 2708

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 850  df-3an 1090  df-tru 1546  df-fal 1556  df-ex 1783  df-sb 2070  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3389  df-v 3430  df-dif 3889  df-un 3891  df-ss 3903  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-mpt 5157  df-iota 6444  df-fv 6496  df-slot 17146

This theorem is referenced by: (None)