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Theorem nfdisj1 5009
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 4996 . 2 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑦∃*𝑥𝐴 𝑦𝐵)
2 nfrmo1 3274 . . 3 𝑥∃*𝑥𝐴 𝑦𝐵
32nfal 2325 . 2 𝑥𝑦∃*𝑥𝐴 𝑦𝐵
41, 3nfxfr 1859 1 𝑥Disj 𝑥𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  wal 1540  wnf 1790  wcel 2114  ∃*wrmo 3056  Disj wdisj 4995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-10 2145  ax-11 2162  ax-12 2179
This theorem depends on definitions:  df-bi 210  df-or 847  df-ex 1787  df-nf 1791  df-mo 2540  df-rmo 3061  df-disj 4996
This theorem is referenced by:  disjabrex  30495  disjabrexf  30496  hasheuni  31623  ldgenpisyslem1  31701  measvunilem  31750  measvunilem0  31751  measvuni  31752  measinblem  31758  voliune  31767  volfiniune  31768  volmeas  31769  dstrvprob  32008  ismeannd  43547
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