Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfabd2OLD Structured version   Visualization version   GIF version

Theorem nfabd2OLD 2971
 Description: Obsolete version of nfabd2 2970 as of 23-May-2023. (Contributed by Mario Carneiro, 8-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
nfabd2.1 𝑦𝜑
nfabd2.2 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfabd2OLD (𝜑𝑥{𝑦𝜓})

Proof of Theorem nfabd2OLD
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfv 1892 . . . 4 𝑧(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
2 df-clab 2776 . . . . 5 (𝑧 ∈ {𝑦𝜓} ↔ [𝑧 / 𝑦]𝜓)
3 nfabd2.1 . . . . . . 7 𝑦𝜑
4 nfnae 2413 . . . . . . 7 𝑦 ¬ ∀𝑥 𝑥 = 𝑦
53, 4nfan 1881 . . . . . 6 𝑦(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
6 nfabd2.2 . . . . . 6 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
75, 6nfsbd 2520 . . . . 5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥[𝑧 / 𝑦]𝜓)
82, 7nfxfrd 1835 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑧 ∈ {𝑦𝜓})
91, 8nfcd 2940 . . 3 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥{𝑦𝜓})
109ex 413 . 2 (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦𝑥{𝑦𝜓}))
11 nfab1 2951 . . 3 𝑦{𝑦𝜓}
12 eqidd 2796 . . . 4 (∀𝑥 𝑥 = 𝑦 → {𝑦𝜓} = {𝑦𝜓})
1312drnfc1 2966 . . 3 (∀𝑥 𝑥 = 𝑦 → (𝑥{𝑦𝜓} ↔ 𝑦{𝑦𝜓}))
1411, 13mpbiri 259 . 2 (∀𝑥 𝑥 = 𝑦𝑥{𝑦𝜓})
1510, 14pm2.61d2 182 1 (𝜑𝑥{𝑦𝜓})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396  ∀wal 1520  Ⅎwnf 1765  [wsb 2042   ∈ wcel 2081  {cab 2775  Ⅎwnfc 2933 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator