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Theorem disjeq 35517
 Description: Equality theorem for disjoints. (Contributed by Peter Mazsa, 22-Sep-2021.)
Assertion
Ref Expression
disjeq (𝐴 = 𝐵 → ( Disj 𝐴 ↔ Disj 𝐵))

Proof of Theorem disjeq
StepHypRef Expression
1 eqimss2 3945 . . 3 (𝐴 = 𝐵𝐵𝐴)
21disjssd 35516 . 2 (𝐴 = 𝐵 → ( Disj 𝐴 → Disj 𝐵))
3 eqimss 3944 . . 3 (𝐴 = 𝐵𝐴𝐵)
43disjssd 35516 . 2 (𝐴 = 𝐵 → ( Disj 𝐵 → Disj 𝐴))
52, 4impbid 213 1 (𝐴 = 𝐵 → ( Disj 𝐴 ↔ Disj 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   = wceq 1522   Disj wdisjALTV 35038 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-br 4963  df-opab 5025  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-coss 35209  df-cnvrefrel 35315  df-funALTV 35465  df-disjALTV 35488 This theorem is referenced by:  disjeqi  35518  disjeqd  35519  disjdmqseqeq1  35520
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