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Theorem nfbidf 2091
Description: An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.) df-nf 1709 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 2090 . . 3 (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
41, 2albid 2089 . . 3 (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))
53, 4imbi12d 334 . 2 (𝜑 → ((∃𝑥𝜓 → ∀𝑥𝜓) ↔ (∃𝑥𝜒 → ∀𝑥𝜒)))
6 df-nf 1709 . 2 (Ⅎ𝑥𝜓 ↔ (∃𝑥𝜓 → ∀𝑥𝜓))
7 df-nf 1709 . 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 1480  wex 1703  wnf 1707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-12 2046
This theorem depends on definitions:  df-bi 197  df-ex 1704  df-nf 1709
This theorem is referenced by:  drnf2  2329  dvelimdf  2334  nfcjust  2751  nfceqdf  2759  bj-drnf2v  32735  bj-nfcjust  32834  wl-nfimf1  33293  nfbii2  33947
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