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

Theorem nfrexd 3344
Description: Deduction version of nfrex 3346. Usage of this theorem is discouraged because it depends on ax-13 2370. See nfrexdw 3291 for a version with a disjoint variable condition, but not requiring ax-13 2370. (Contributed by Mario Carneiro, 14-Oct-2016.) (New usage is discouraged.)
Hypotheses
Ref Expression
nfrald.1 𝑦𝜑
nfrald.2 (𝜑𝑥𝐴)
nfrald.3 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfrexd (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)

Proof of Theorem nfrexd
StepHypRef Expression
1 dfrex2 3074 . 2 (∃𝑦𝐴 𝜓 ↔ ¬ ∀𝑦𝐴 ¬ 𝜓)
2 nfrald.1 . . . 4 𝑦𝜑
3 nfrald.2 . . . 4 (𝜑𝑥𝐴)
4 nfrald.3 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
54nfnd 1861 . . . 4 (𝜑 → Ⅎ𝑥 ¬ 𝜓)
62, 3, 5nfrald 3343 . . 3 (𝜑 → Ⅎ𝑥𝑦𝐴 ¬ 𝜓)
76nfnd 1861 . 2 (𝜑 → Ⅎ𝑥 ¬ ∀𝑦𝐴 ¬ 𝜓)
81, 7nfxfrd 1856 1 (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wnf 1785  wnfc 2885  wral 3062  wrex 3071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2370  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ral 3063  df-rex 3072
This theorem is referenced by:  nfrex  3346  nfiundg  47052
  Copyright terms: Public domain W3C validator