Theorem disjeqi 38211

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 38210	. 2 ⊢ (𝐴 = 𝐵 → ( Disj 𝐴 ↔ Disj 𝐵))
3	1, 2	ax-mp 5	1 ⊢ ( Disj 𝐴 ↔ Disj 𝐵)

Syntax hints:   ↔ wb 205   = wceq 1533   Disj wdisjALTV 37687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5151  df-opab 5213  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-coss 37887  df-cnvrefrel 38003  df-funALTV 38158  df-disjALTV 38181

This theorem is referenced by:  disjxrnres5  38223  disjsuc  38235