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Theorem nfbii 1362
 Description: Equality theorem for not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
Hypothesis
Ref Expression
nfbii.1 (𝜑𝜓)
Assertion
Ref Expression
nfbii (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓)

Proof of Theorem nfbii
StepHypRef Expression
1 nfbii.1 . . . 4 (𝜑𝜓)
21albii 1359 . . . 4 (∀𝑥𝜑 ↔ ∀𝑥𝜓)
31, 2imbi12i 228 . . 3 ((𝜑 → ∀𝑥𝜑) ↔ (𝜓 → ∀𝑥𝜓))
43albii 1359 . 2 (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥(𝜓 → ∀𝑥𝜓))
5 df-nf 1350 . 2 (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
6 df-nf 1350 . 2 (Ⅎ𝑥𝜓 ↔ ∀𝑥(𝜓 → ∀𝑥𝜓))
74, 5, 63bitr4i 201 1 (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241  Ⅎwnf 1349 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338 This theorem depends on definitions:  df-bi 110  df-nf 1350 This theorem is referenced by:  nfxfr  1363  nfxfrd  1364  nfsb  1822  nfsbt  1850  hbsbd  1858  sbal1yz  1877  dvelimALT  1886  dvelimfv  1887  dvelimor  1894  nfeudv  1915  nfeuv  1918  nfceqi  2174  nfreudxy  2483  dfnfc2  3598
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