Theorem disjeqi 38640

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 38639	. 2 ⊢ (𝐴 = 𝐵 → ( Disj 𝐴 ↔ Disj 𝐵))
3	1, 2	ax-mp 5	1 ⊢ ( Disj 𝐴 ↔ Disj 𝐵)

Syntax hints:   ↔ wb 206   = wceq 1537   Disj wdisjALTV 38118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5170  df-opab 5232  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-coss 38316  df-cnvrefrel 38432  df-funALTV 38587  df-disjALTV 38610

This theorem is referenced by:  disjxrnres5  38652  disjsuc  38664