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Theorem nfdisj 4740
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 4730 . 2 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑧∃*𝑥(𝑥𝐴𝑧𝐵))
2 nftru 1843 . . . . 5 𝑥
3 nfcvf 2890 . . . . . . . 8 (¬ ∀𝑦 𝑦 = 𝑥𝑦𝑥)
4 nfdisj.1 . . . . . . . . 9 𝑦𝐴
54a1i 11 . . . . . . . 8 (¬ ∀𝑦 𝑦 = 𝑥𝑦𝐴)
63, 5nfeld 2875 . . . . . . 7 (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 𝑥𝐴)
7 nfdisj.2 . . . . . . . . 9 𝑦𝐵
87nfcri 2860 . . . . . . . 8 𝑦 𝑧𝐵
98a1i 11 . . . . . . 7 (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 𝑧𝐵)
106, 9nfand 1939 . . . . . 6 (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦(𝑥𝐴𝑧𝐵))
1110adantl 473 . . . . 5 ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦(𝑥𝐴𝑧𝐵))
122, 11nfmod2 2584 . . . 4 (⊤ → Ⅎ𝑦∃*𝑥(𝑥𝐴𝑧𝐵))
1312trud 1606 . . 3 𝑦∃*𝑥(𝑥𝐴𝑧𝐵)
1413nfal 2264 . 2 𝑦𝑧∃*𝑥(𝑥𝐴𝑧𝐵)
151, 14nfxfr 1892 1 𝑦Disj 𝑥𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 383  wal 1594  wtru 1597  wnf 1821  wcel 2103  ∃*wmo 2572  wnfc 2853  Disj wdisj 4728
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-cleq 2717  df-clel 2720  df-nfc 2855  df-rmo 3022  df-disj 4729
This theorem is referenced by:  disjxiun  4757
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