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

Theorem nfbii 1770
 Description: Equality theorem for not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1701 changed. (Revised by Wolf Lammen, 12-Sep-2021.)
Hypothesis
Ref Expression
nfbii.1 (𝜑𝜓)
Assertion
Ref Expression
nfbii (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓)

Proof of Theorem nfbii
StepHypRef Expression
1 nfbii.1 . . . 4 (𝜑𝜓)
21exbii 1764 . . 3 (∃𝑥𝜑 ↔ ∃𝑥𝜓)
31albii 1737 . . 3 (∀𝑥𝜑 ↔ ∀𝑥𝜓)
42, 3imbi12i 339 . 2 ((∃𝑥𝜑 → ∀𝑥𝜑) ↔ (∃𝑥𝜓 → ∀𝑥𝜓))
5 df-nf 1701 . 2 (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
6 df-nf 1701 . 2 (Ⅎ𝑥𝜓 ↔ (∃𝑥𝜓 → ∀𝑥𝜓))
74, 5, 63bitr4i 291 1 (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473  ∃wex 1695  Ⅎwnf 1699 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728 This theorem depends on definitions:  df-bi 196  df-ex 1696  df-nf 1701 This theorem is referenced by:  nfxfr  1771  nfxfrd  1772  dvelimhw  2159  nfeqf1  2287  dfnfc2  4390  dfnfc2OLD  4391  bj-dvelimdv1  32028  bj-nfcf  32112  iunconlem2  38193
 Copyright terms: Public domain W3C validator