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Theorem nfdisj1 5127
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 5114 . 2 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑦∃*𝑥𝐴 𝑦𝐵)
2 nfrmo1 3407 . . 3 𝑥∃*𝑥𝐴 𝑦𝐵
32nfal 2316 . 2 𝑥𝑦∃*𝑥𝐴 𝑦𝐵
41, 3nfxfr 1855 1 𝑥Disj 𝑥𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  wal 1539  wnf 1785  wcel 2106  ∃*wrmo 3375  Disj wdisj 5113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 846  df-ex 1782  df-nf 1786  df-mo 2534  df-rmo 3376  df-disj 5114
This theorem is referenced by:  disjabrex  32068  disjabrexf  32069  hasheuni  33369  ldgenpisyslem1  33447  measvunilem  33496  measvunilem0  33497  measvuni  33498  measinblem  33504  voliune  33513  volfiniune  33514  volmeas  33515  dstrvprob  33756  ismeannd  45482
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