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Theorem nfdisj1 5129
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 5116 . 2 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑦∃*𝑥𝐴 𝑦𝐵)
2 nfrmo1 3409 . . 3 𝑥∃*𝑥𝐴 𝑦𝐵
32nfal 2322 . 2 𝑥𝑦∃*𝑥𝐴 𝑦𝐵
41, 3nfxfr 1850 1 𝑥Disj 𝑥𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  wal 1535  wnf 1780  wcel 2106  ∃*wrmo 3377  Disj wdisj 5115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-10 2139  ax-11 2155  ax-12 2175
This theorem depends on definitions:  df-bi 207  df-or 848  df-ex 1777  df-nf 1781  df-mo 2538  df-rmo 3378  df-disj 5116
This theorem is referenced by:  disjabrex  32602  disjabrexf  32603  hasheuni  34066  ldgenpisyslem1  34144  measvunilem  34193  measvunilem0  34194  measvuni  34195  measinblem  34201  voliune  34210  volfiniune  34211  volmeas  34212  dstrvprob  34453  ismeannd  46423
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