Theorem disjeqi 39591

Description: Equality theorem for disjoints, inference version. (Contributed by Peter Mazsa, 22-Sep-2021.)

Hypothesis

Ref	Expression
disjeqi.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
disjeqi	⊢ ( Disj 𝐴 ↔ Disj 𝐵)

Proof of Theorem disjeqi

Step	Hyp	Ref	Expression
1		disjeqi.1	. 2 ⊢ 𝐴 = 𝐵
2		disjeq 39590	. 2 ⊢ (𝐴 = 𝐵 → ( Disj 𝐴 ↔ Disj 𝐵))
3	1, 2	ax-mp 5	1 ⊢ ( Disj 𝐴 ↔ Disj 𝐵)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   Disj wdisjALTV 38975

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 2147  ax-9 2155  ax-11 2194  ax-ext 2734  ax-sep 5255  ax-pr 5402

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 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-coss 39257  df-cnvrefrel 39363  df-funALTV 39523  df-disjALTV 39546

This theorem is used by:  disjxrnres5  39603  disjsuc  39615