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Theorem r19.35 3065
 Description: Restricted quantifier version of 19.35 1794. (Contributed by NM, 20-Sep-2003.)
Assertion
Ref Expression
r19.35 (∃𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))

Proof of Theorem r19.35
StepHypRef Expression
1 r19.26 3046 . . . 4 (∀𝑥𝐴 (𝜑 ∧ ¬ 𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 ¬ 𝜓))
2 annim 440 . . . . 5 ((𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑𝜓))
32ralbii 2963 . . . 4 (∀𝑥𝐴 (𝜑 ∧ ¬ 𝜓) ↔ ∀𝑥𝐴 ¬ (𝜑𝜓))
4 df-an 385 . . . 4 ((∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 ¬ 𝜓) ↔ ¬ (∀𝑥𝐴 𝜑 → ¬ ∀𝑥𝐴 ¬ 𝜓))
51, 3, 43bitr3i 289 . . 3 (∀𝑥𝐴 ¬ (𝜑𝜓) ↔ ¬ (∀𝑥𝐴 𝜑 → ¬ ∀𝑥𝐴 ¬ 𝜓))
65con2bii 346 . 2 ((∀𝑥𝐴 𝜑 → ¬ ∀𝑥𝐴 ¬ 𝜓) ↔ ¬ ∀𝑥𝐴 ¬ (𝜑𝜓))
7 dfrex2 2979 . . 3 (∃𝑥𝐴 𝜓 ↔ ¬ ∀𝑥𝐴 ¬ 𝜓)
87imbi2i 325 . 2 ((∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓) ↔ (∀𝑥𝐴 𝜑 → ¬ ∀𝑥𝐴 ¬ 𝜓))
9 dfrex2 2979 . 2 (∃𝑥𝐴 (𝜑𝜓) ↔ ¬ ∀𝑥𝐴 ¬ (𝜑𝜓))
106, 8, 93bitr4ri 292 1 (∃𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  ∀wral 2896  ∃wrex 2897 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-ral 2901  df-rex 2902 This theorem is referenced by:  r19.36v  3066  r19.37  3067  r19.43  3074  r19.37zv  4019  r19.36zv  4024  iinexg  4751  bndndx  11168  nmobndseqi  27018  nmobndseqiALT  27019
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