MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfan1OLD Structured version   Visualization version   GIF version

Theorem nfan1OLD 2272
Description: Obsolete proof of nfan1 2106 as of 6-Oct-2021. (Contributed by Mario Carneiro, 3-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
nfan1OLD.1 𝑥𝜑
nfan1OLD.2 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfan1OLD 𝑥(𝜑𝜓)

Proof of Theorem nfan1OLD
StepHypRef Expression
1 nfan1OLD.2 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
21nfrdOLD 2226 . . . 4 (𝜑 → (𝜓 → ∀𝑥𝜓))
32imdistani 726 . . 3 ((𝜑𝜓) → (𝜑 ∧ ∀𝑥𝜓))
4 nfan1OLD.1 . . . 4 𝑥𝜑
5419.28OLD 2271 . . 3 (∀𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))
63, 5sylibr 224 . 2 ((𝜑𝜓) → ∀𝑥(𝜑𝜓))
76nfiOLD 1774 1 𝑥(𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1521  wnfOLD 1749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-12 2087
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-nfOLD 1761
This theorem is referenced by:  nfanOLDOLD  2273
  Copyright terms: Public domain W3C validator