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Theorem nfbidf 2203
Description: An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.) df-nf 1823 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
Hypotheses
Ref Expression
albid.1 𝑥𝜑
albid.2 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
nfbidf (𝜑 → (Ⅎ𝑥𝜓 ↔ Ⅎ𝑥𝜒))

Proof of Theorem nfbidf
StepHypRef Expression
1 albid.1 . . . 4 𝑥𝜑
2 albid.2 . . . 4 (𝜑 → (𝜓𝜒))
31, 2exbid 2202 . . 3 (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
41, 2albid 2201 . . 3 (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))
53, 4imbi12d 333 . 2 (𝜑 → ((∃𝑥𝜓 → ∀𝑥𝜓) ↔ (∃𝑥𝜒 → ∀𝑥𝜒)))
6 df-nf 1823 . 2 (Ⅎ𝑥𝜓 ↔ (∃𝑥𝜓 → ∀𝑥𝜓))
7 df-nf 1823 . 2 (Ⅎ𝑥𝜒 ↔ (∃𝑥𝜒 → ∀𝑥𝜒))
85, 6, 73bitr4g 303 1 (𝜑 → (Ⅎ𝑥𝜓 ↔ Ⅎ𝑥𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1594  wex 1817  wnf 1821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-12 2160
This theorem depends on definitions:  df-bi 197  df-ex 1818  df-nf 1823
This theorem is referenced by:  drnf2  2434  dvelimdf  2439  nfcjust  2854  nfceqdf  2862  bj-drnf2v  32978  bj-nfcjust  33077  wl-nfimf1  33545  nfbii2  34199
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