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Theorem disjeq2dv 5111
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 3140 . 2 (𝜑 → ∀𝑥𝐴 𝐵 = 𝐶)
3 disjeq2 5110 . 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 205  wa 395   = wceq 1533  wcel 2098  wral 3055  Disj wdisj 5106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-mo 2528  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rmo 3370  df-v 3470  df-in 3950  df-ss 3960  df-disj 5107
This theorem is referenced by:  disjeq12d  5115  iunmbl  25437  uniioovol  25463  tocyccntz  32809  carsggect  33847  voliunnfl  37045  nnfoctbdjlem  45743  meadjiun  45754
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