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Theorem disjeq2dv 4782
Description: Equality deduction for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Hypothesis
Ref Expression
disjeq2dv.1 ((𝜑𝑥𝐴) → 𝐵 = 𝐶)
Assertion
Ref Expression
disjeq2dv (𝜑 → (Disj 𝑥𝐴 𝐵Disj 𝑥𝐴 𝐶))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem disjeq2dv
StepHypRef Expression
1 disjeq2dv.1 . . 3 ((𝜑𝑥𝐴) → 𝐵 = 𝐶)
21ralrimiva 3113 . 2 (𝜑 → ∀𝑥𝐴 𝐵 = 𝐶)
3 disjeq2 4781 . 2 (∀𝑥𝐴 𝐵 = 𝐶 → (Disj 𝑥𝐴 𝐵Disj 𝑥𝐴 𝐶))
42, 3syl 17 1 (𝜑 → (Disj 𝑥𝐴 𝐵Disj 𝑥𝐴 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wral 3055  Disj wdisj 4777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-clab 2752  df-cleq 2758  df-clel 2761  df-ral 3060  df-rmo 3063  df-in 3739  df-ss 3746  df-disj 4778
This theorem is referenced by:  disjeq12d  4786  iunmbl  23611  uniioovol  23637  carsggect  30762  voliunnfl  33809  nnfoctbdjlem  41241  meadjiun  41252
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