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Theorem r19.35 3077
Description: Restricted quantifier version of 19.35 1802. (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 3058 . . . 4 (∀𝑥𝐴 (𝜑 ∧ ¬ 𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 ¬ 𝜓))
2 annim 441 . . . . 5 ((𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑𝜓))
32ralbii 2975 . . . 4 (∀𝑥𝐴 (𝜑 ∧ ¬ 𝜓) ↔ ∀𝑥𝐴 ¬ (𝜑𝜓))
4 df-an 386 . . . 4 ((∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 ¬ 𝜓) ↔ ¬ (∀𝑥𝐴 𝜑 → ¬ ∀𝑥𝐴 ¬ 𝜓))
51, 3, 43bitr3i 290 . . 3 (∀𝑥𝐴 ¬ (𝜑𝜓) ↔ ¬ (∀𝑥𝐴 𝜑 → ¬ ∀𝑥𝐴 ¬ 𝜓))
65con2bii 347 . 2 ((∀𝑥𝐴 𝜑 → ¬ ∀𝑥𝐴 ¬ 𝜓) ↔ ¬ ∀𝑥𝐴 ¬ (𝜑𝜓))
7 dfrex2 2991 . . 3 (∃𝑥𝐴 𝜓 ↔ ¬ ∀𝑥𝐴 ¬ 𝜓)
87imbi2i 326 . 2 ((∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓) ↔ (∀𝑥𝐴 𝜑 → ¬ ∀𝑥𝐴 ¬ 𝜓))
9 dfrex2 2991 . 2 (∃𝑥𝐴 (𝜑𝜓) ↔ ¬ ∀𝑥𝐴 ¬ (𝜑𝜓))
106, 8, 93bitr4ri 293 1 (∃𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  wral 2907  wrex 2908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-ral 2912  df-rex 2913
This theorem is referenced by:  r19.36v  3078  r19.37  3079  r19.43  3086  r19.37zv  4044  r19.36zv  4049  iinexg  4789  bndndx  11243  nmobndseqi  27504  nmobndseqiALT  27505  r19.36vf  38846
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