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Theorem nfbidv 1923
Description: An equality theorem for nonfreeness. See nfbidf 2225 for a version without disjoint variable condition but requiring more axioms. (Contributed by Mario Carneiro, 4-Oct-2016.) Remove dependency on ax-6 1970, ax-7 2015, ax-12 2176 by adapting proof of nfbidf 2225. (Revised by BJ, 25-Sep-2022.)
Hypothesis
Ref Expression
albidv.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
nfbidv (𝜑 → (Ⅎ𝑥𝜓 ↔ Ⅎ𝑥𝜒))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)

Proof of Theorem nfbidv
StepHypRef Expression
1 albidv.1 . . . 4 (𝜑 → (𝜓𝜒))
21exbidv 1922 . . 3 (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
31albidv 1921 . . 3 (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))
42, 3imbi12d 348 . 2 (𝜑 → ((∃𝑥𝜓 → ∀𝑥𝜓) ↔ (∃𝑥𝜒 → ∀𝑥𝜒)))
5 df-nf 1786 . 2 (Ⅎ𝑥𝜓 ↔ (∃𝑥𝜓 → ∀𝑥𝜓))
6 df-nf 1786 . 2 (Ⅎ𝑥𝜒 ↔ (∃𝑥𝜒 → ∀𝑥𝜒))
74, 5, 63bitr4g 317 1 (𝜑 → (Ⅎ𝑥𝜓 ↔ Ⅎ𝑥𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1536  wex 1781  wnf 1785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911
This theorem depends on definitions:  df-bi 210  df-ex 1782  df-nf 1786
This theorem is referenced by:  nfcjust  2940  nfcr  2944  nfcriOLD  2949  nfcriOLDOLD  2950  bj-drnf2v  34242
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