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Theorem nfdisj 5040
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 5029 . 2 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑧∃*𝑥(𝑥𝐴𝑧𝐵))
2 nftru 1798 . . . . 5 𝑥
3 nfcvf 3011 . . . . . . . 8 (¬ ∀𝑦 𝑦 = 𝑥𝑦𝑥)
4 nfdisj.1 . . . . . . . . 9 𝑦𝐴
54a1i 11 . . . . . . . 8 (¬ ∀𝑦 𝑦 = 𝑥𝑦𝐴)
63, 5nfeld 2993 . . . . . . 7 (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 𝑥𝐴)
7 nfdisj.2 . . . . . . . . 9 𝑦𝐵
87nfcri 2975 . . . . . . . 8 𝑦 𝑧𝐵
98a1i 11 . . . . . . 7 (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 𝑧𝐵)
106, 9nfand 1891 . . . . . 6 (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦(𝑥𝐴𝑧𝐵))
1110adantl 482 . . . . 5 ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦(𝑥𝐴𝑧𝐵))
122, 11nfmod2 2639 . . . 4 (⊤ → Ⅎ𝑦∃*𝑥(𝑥𝐴𝑧𝐵))
1312mptru 1537 . . 3 𝑦∃*𝑥(𝑥𝐴𝑧𝐵)
1413nfal 2336 . 2 𝑦𝑧∃*𝑥(𝑥𝐴𝑧𝐵)
151, 14nfxfr 1846 1 𝑦Disj 𝑥𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396  wal 1528  wtru 1531  wnf 1777  wcel 2107  ∃*wmo 2617  wnfc 2965  Disj wdisj 5027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2797
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-cleq 2818  df-clel 2897  df-nfc 2967  df-rmo 3150  df-disj 5028
This theorem is referenced by: (None)
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