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Theorem nfdisj1 5053
Description: Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
nfdisj1 𝑥Disj 𝑥𝐴 𝐵

Proof of Theorem nfdisj1
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-disj 5040 . 2 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑦∃*𝑥𝐴 𝑦𝐵)
2 nfrmo1 3301 . . 3 𝑥∃*𝑥𝐴 𝑦𝐵
32nfal 2317 . 2 𝑥𝑦∃*𝑥𝐴 𝑦𝐵
41, 3nfxfr 1855 1 𝑥Disj 𝑥𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  wal 1537  wnf 1786  wcel 2106  ∃*wrmo 3067  Disj wdisj 5039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-11 2154  ax-12 2171
This theorem depends on definitions:  df-bi 206  df-or 845  df-ex 1783  df-nf 1787  df-mo 2540  df-rmo 3071  df-disj 5040
This theorem is referenced by:  disjabrex  30921  disjabrexf  30922  hasheuni  32053  ldgenpisyslem1  32131  measvunilem  32180  measvunilem0  32181  measvuni  32182  measinblem  32188  voliune  32197  volfiniune  32198  volmeas  32199  dstrvprob  32438  ismeannd  44005
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