Theorem disjeqi 39346

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 39345	. 2 ⊢ (𝐴 = 𝐵 → ( Disj 𝐴 ↔ Disj 𝐵))
3	1, 2	ax-mp 5	1 ⊢ ( Disj 𝐴 ↔ Disj 𝐵)

Syntax hints:   ↔ wb 209   = wceq 1563   Disj wdisjALTV 38730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-11 2194  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-coss 39012  df-cnvrefrel 39118  df-funALTV 39278  df-disjALTV 39301

This theorem is referenced by:  disjxrnres5  39358  disjsuc  39370