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

Theorem nfreud 3284
Description: Deduction version of nfreu 3288. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 8-Oct-2016.) (New usage is discouraged.)
Hypotheses
Ref Expression
nfreud.1 𝑦𝜑
nfreud.2 (𝜑𝑥𝐴)
nfreud.3 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfreud (𝜑 → Ⅎ𝑥∃!𝑦𝐴 𝜓)

Proof of Theorem nfreud
StepHypRef Expression
1 df-reu 3068 . 2 (∃!𝑦𝐴 𝜓 ↔ ∃!𝑦(𝑦𝐴𝜓))
2 nfreud.1 . . 3 𝑦𝜑
3 nfcvf 2933 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
43adantl 485 . . . . 5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥𝑦)
5 nfreud.2 . . . . . 6 (𝜑𝑥𝐴)
65adantr 484 . . . . 5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥𝐴)
74, 6nfeld 2915 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦𝐴)
8 nfreud.3 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
98adantr 484 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
107, 9nfand 1905 . . 3 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦𝐴𝜓))
112, 10nfeud2 2589 . 2 (𝜑 → Ⅎ𝑥∃!𝑦(𝑦𝐴𝜓))
121, 11nfxfrd 1861 1 (𝜑 → Ⅎ𝑥∃!𝑦𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wal 1541  wnf 1791  wcel 2110  ∃!weu 2567  wnfc 2884  ∃!wreu 3063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-13 2371  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-mo 2539  df-eu 2568  df-cleq 2729  df-clel 2816  df-nfc 2886  df-reu 3068
This theorem is referenced by:  nfreu  3288
  Copyright terms: Public domain W3C validator