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Theorem disjeq1f 32573
Description: Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Hypotheses
Ref Expression
disjss1f.1 𝑥𝐴
disjss1f.2 𝑥𝐵
Assertion
Ref Expression
disjeq1f (𝐴 = 𝐵 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))

Proof of Theorem disjeq1f
StepHypRef Expression
1 eqimss2 4055 . . 3 (𝐴 = 𝐵𝐵𝐴)
2 disjss1f.2 . . . 4 𝑥𝐵
3 disjss1f.1 . . . 4 𝑥𝐴
42, 3disjss1f 32572 . . 3 (𝐵𝐴 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
51, 4syl 17 . 2 (𝐴 = 𝐵 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
6 eqimss 4054 . . 3 (𝐴 = 𝐵𝐴𝐵)
73, 2disjss1f 32572 . . 3 (𝐴𝐵 → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐴 𝐶))
86, 7syl 17 . 2 (𝐴 = 𝐵 → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐴 𝐶))
95, 8impbid 212 1 (𝐴 = 𝐵 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1535  wnfc 2886  wss 3963  Disj wdisj 5116
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-11 2153  ax-12 2173  ax-ext 2704
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1775  df-nf 1779  df-mo 2536  df-cleq 2725  df-clel 2812  df-nfc 2888  df-rmo 3376  df-ss 3980  df-disj 5117
This theorem is referenced by:  ldgenpisyslem1  34105
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