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Theorem nfdisj1 5124
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 5111 . 2 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑦∃*𝑥𝐴 𝑦𝐵)
2 nfrmo1 3411 . . 3 𝑥∃*𝑥𝐴 𝑦𝐵
32nfal 2323 . 2 𝑥𝑦∃*𝑥𝐴 𝑦𝐵
41, 3nfxfr 1853 1 𝑥Disj 𝑥𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  wal 1538  wnf 1783  wcel 2108  ∃*wrmo 3379  Disj wdisj 5110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-10 2141  ax-11 2157  ax-12 2177
This theorem depends on definitions:  df-bi 207  df-or 849  df-ex 1780  df-nf 1784  df-mo 2540  df-rmo 3380  df-disj 5111
This theorem is referenced by:  disjabrex  32595  disjabrexf  32596  hasheuni  34086  ldgenpisyslem1  34164  measvunilem  34213  measvunilem0  34214  measvuni  34215  measinblem  34221  voliune  34230  volfiniune  34231  volmeas  34232  dstrvprob  34474  ismeannd  46482
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