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

Theorem nfdisj 5035
Description: Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Hypotheses
Ref Expression
nfdisj.1 𝑦𝐴
nfdisj.2 𝑦𝐵
Assertion
Ref Expression
nfdisj 𝑦Disj 𝑥𝐴 𝐵

Proof of Theorem nfdisj
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dfdisj2 5024 . 2 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑧∃*𝑥(𝑥𝐴𝑧𝐵))
2 nftru 1796 . . . . 5 𝑥
3 nfcvf 3004 . . . . . . . 8 (¬ ∀𝑦 𝑦 = 𝑥𝑦𝑥)
4 nfdisj.1 . . . . . . . . 9 𝑦𝐴
54a1i 11 . . . . . . . 8 (¬ ∀𝑦 𝑦 = 𝑥𝑦𝐴)
63, 5nfeld 2986 . . . . . . 7 (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 𝑥𝐴)
7 nfdisj.2 . . . . . . . . 9 𝑦𝐵
87nfcri 2968 . . . . . . . 8 𝑦 𝑧𝐵
98a1i 11 . . . . . . 7 (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 𝑧𝐵)
106, 9nfand 1889 . . . . . 6 (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦(𝑥𝐴𝑧𝐵))
1110adantl 482 . . . . 5 ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦(𝑥𝐴𝑧𝐵))
122, 11nfmod2 2635 . . . 4 (⊤ → Ⅎ𝑦∃*𝑥(𝑥𝐴𝑧𝐵))
1312mptru 1535 . . 3 𝑦∃*𝑥(𝑥𝐴𝑧𝐵)
1413nfal 2333 . 2 𝑦𝑧∃*𝑥(𝑥𝐴𝑧𝐵)
151, 14nfxfr 1844 1 𝑦Disj 𝑥𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396  wal 1526  wtru 1529  wnf 1775  wcel 2105  ∃*wmo 2613  wnfc 2958  Disj wdisj 5022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-13 2381  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rmo 3143  df-disj 5023
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator