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Theorem nfbii 1403
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 1400 . . . 4 (∀𝑥𝜑 ↔ ∀𝑥𝜓)
31, 2imbi12i 237 . . 3 ((𝜑 → ∀𝑥𝜑) ↔ (𝜓 → ∀𝑥𝜓))
43albii 1400 . 2 (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥(𝜓 → ∀𝑥𝜓))
5 df-nf 1391 . 2 (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
6 df-nf 1391 . 2 (Ⅎ𝑥𝜓 ↔ ∀𝑥(𝜓 → ∀𝑥𝜓))
74, 5, 63bitr4i 210 1 (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1283  wnf 1390
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379
This theorem depends on definitions:  df-bi 115  df-nf 1391
This theorem is referenced by:  nfxfr  1404  nfxfrd  1405  nfsb  1865  nfsbt  1893  hbsbd  1901  sbal1yz  1920  dvelimALT  1929  dvelimfv  1930  dvelimor  1937  nfeudv  1958  nfeuv  1961  nfceqi  2219  nfreudxy  2532  dfnfc2  3639
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