Theorem sloteq 13077

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 5635	. . 3 ⊢ (𝐴 = 𝐵 → (𝑓‘𝐴) = (𝑓‘𝐵))
2	1	mpteq2dv 4178	. 2 ⊢ (𝐴 = 𝐵 → (𝑓 ∈ V ↦ (𝑓‘𝐴)) = (𝑓 ∈ V ↦ (𝑓‘𝐵)))
3		df-slot 13076	. 2 ⊢ Slot 𝐴 = (𝑓 ∈ V ↦ (𝑓‘𝐴))
4		df-slot 13076	. 2 ⊢ Slot 𝐵 = (𝑓 ∈ V ↦ (𝑓‘𝐵))
5	2, 3, 4	3eqtr4g 2287	1 ⊢ (𝐴 = 𝐵 → Slot 𝐴 = Slot 𝐵)

Syntax hints:   → wi 4   = wceq 1395  Vcvv 2800   ↦ cmpt 4148  ‘cfv 5324  Slot cslot 13071

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-mpt 4150  df-iota 5284  df-fv 5332  df-slot 13076

This theorem is referenced by:  ndxid  13096