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Theorem disjeq1d 4780
Description: Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Hypothesis
Ref Expression
disjeq1d.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
disjeq1d (𝜑 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝐶(𝑥)

Proof of Theorem disjeq1d
StepHypRef Expression
1 disjeq1d.1 . 2 (𝜑𝐴 = 𝐵)
2 disjeq1 4779 . 2 (𝐴 = 𝐵 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
31, 2syl 17 1 (𝜑 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1632  Disj wdisj 4772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-rmo 3058  df-in 3722  df-ss 3729  df-disj 4773
This theorem is referenced by:  disjeq12d  4781  disjxiun  4801  disjdifprg  29695  disjdifprg2  29696  disjun0  29715  measxun2  30582  measssd  30587  meadjun  41182
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