Theorem sloteq 17261

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.

Step	Hyp	Ref	Expression
1		fveq2 6885	. . 3 ⊢ (𝐴 = 𝐵 → (𝑓‘𝐴) = (𝑓‘𝐵))
2	1	mpteq2dv 5207	. 2 ⊢ (𝐴 = 𝐵 → (𝑓 ∈ V ↦ (𝑓‘𝐴)) = (𝑓 ∈ V ↦ (𝑓‘𝐵)))
3		df-slot 17260	. 2 ⊢ Slot 𝐴 = (𝑓 ∈ V ↦ (𝑓‘𝐴))
4		df-slot 17260	. 2 ⊢ Slot 𝐵 = (𝑓 ∈ V ↦ (𝑓‘𝐵))
5	2, 3, 4	3eqtr4g 2825	1 ⊢ (𝐴 = 𝐵 → Slot 𝐴 = Slot 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1570  Vcvv 3457   ↦ cmpt 5194  ‘cfv 6540  Slot cslot 17259

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-iota 6496  df-fv 6548  df-slot 17260

This theorem is used by:  ndxid  17275