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Theorem nfdisj 5123
Description: Bound-variable hypothesis builder for disjoint collection. Usage of this theorem is discouraged because it depends on ax-13 2377. Use the weaker nfdisjw 5122 when possible. (Contributed by Mario Carneiro, 14-Nov-2016.) (New usage is discouraged.)
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 5112 . 2 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑧∃*𝑥(𝑥𝐴𝑧𝐵))
2 nftru 1804 . . . . 5 𝑥
3 nfcvf 2932 . . . . . . . 8 (¬ ∀𝑦 𝑦 = 𝑥𝑦𝑥)
4 nfdisj.1 . . . . . . . . 9 𝑦𝐴
54a1i 11 . . . . . . . 8 (¬ ∀𝑦 𝑦 = 𝑥𝑦𝐴)
63, 5nfeld 2917 . . . . . . 7 (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 𝑥𝐴)
7 nfdisj.2 . . . . . . . . 9 𝑦𝐵
87nfcri 2897 . . . . . . . 8 𝑦 𝑧𝐵
98a1i 11 . . . . . . 7 (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 𝑧𝐵)
106, 9nfand 1897 . . . . . 6 (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦(𝑥𝐴𝑧𝐵))
1110adantl 481 . . . . 5 ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦(𝑥𝐴𝑧𝐵))
122, 11nfmod2 2558 . . . 4 (⊤ → Ⅎ𝑦∃*𝑥(𝑥𝐴𝑧𝐵))
1312mptru 1547 . . 3 𝑦∃*𝑥(𝑥𝐴𝑧𝐵)
1413nfal 2323 . 2 𝑦𝑧∃*𝑥(𝑥𝐴𝑧𝐵)
151, 14nfxfr 1853 1 𝑦Disj 𝑥𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395  wal 1538  wtru 1541  wnf 1783  wcel 2108  ∃*wmo 2538  wnfc 2890  Disj wdisj 5110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-13 2377  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-mo 2540  df-cleq 2729  df-clel 2816  df-nfc 2892  df-rmo 3380  df-disj 5111
This theorem is referenced by: (None)
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