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Theorem nfbidf 2221
 Description: An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.) df-nf 1781 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 2220 . . 3 (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
41, 2albid 2219 . . 3 (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))
53, 4imbi12d 347 . 2 (𝜑 → ((∃𝑥𝜓 → ∀𝑥𝜓) ↔ (∃𝑥𝜒 → ∀𝑥𝜒)))
6 df-nf 1781 . 2 (Ⅎ𝑥𝜓 ↔ (∃𝑥𝜓 → ∀𝑥𝜓))
7 df-nf 1781 . 2 (Ⅎ𝑥𝜒 ↔ (∃𝑥𝜒 → ∀𝑥𝜒))
85, 6, 73bitr4g 316 1 (𝜑 → (Ⅎ𝑥𝜓 ↔ Ⅎ𝑥𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1531  ∃wex 1776  Ⅎwnf 1780 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-12 2172 This theorem depends on definitions:  df-bi 209  df-ex 1777  df-nf 1781 This theorem is referenced by:  drnf2  2462  dvelimdf  2467  nfceqdf  2972  nfabdw  3000  wl-nfimf1  34760  ichnfimlem2  43615
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