Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfdisj1 Structured version   Visualization version   GIF version

Theorem nfdisj1 4785
 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 4773 . 2 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑦∃*𝑥𝐴 𝑦𝐵)
2 nfrmo1 3249 . . 3 𝑥∃*𝑥𝐴 𝑦𝐵
32nfal 2300 . 2 𝑥𝑦∃*𝑥𝐴 𝑦𝐵
41, 3nfxfr 1928 1 𝑥Disj 𝑥𝐴 𝐵
 Colors of variables: wff setvar class Syntax hints:  ∀wal 1630  Ⅎwnf 1857   ∈ wcel 2139  ∃*wrmo 3053  Disj wdisj 4772 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-10 2168  ax-11 2183  ax-12 2196 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1854  df-nf 1859  df-eu 2611  df-mo 2612  df-rmo 3058  df-disj 4773 This theorem is referenced by:  disjabrex  29702  disjabrexf  29703  hasheuni  30456  ldgenpisyslem1  30535  measvunilem  30584  measvunilem0  30585  measvuni  30586  measinblem  30592  voliune  30601  volfiniune  30602  volmeas  30603  dstrvprob  30842  ismeannd  41187
 Copyright terms: Public domain W3C validator