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Theorem disjeq1d 4551
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 4550 . 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 194   = wceq 1474  Disj wdisj 4543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-rmo 2899  df-in 3542  df-ss 3549  df-disj 4544
This theorem is referenced by:  disjeq12d  4552  disjxiun  4569  disjxiunOLD  4570  disjdifprg  28572  disjdifprg2  28573  disjun0  28592  measxun2  29402  measssd  29407  meadjun  39155
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