MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  r19.35 Structured version   Visualization version   GIF version

Theorem r19.35 3231
Description: Restricted quantifier version of 19.35 1956. (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 3211 . . . 4 (∀𝑥𝐴 (𝜑 ∧ ¬ 𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 ¬ 𝜓))
2 annim 390 . . . . 5 ((𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑𝜓))
32ralbii 3128 . . . 4 (∀𝑥𝐴 (𝜑 ∧ ¬ 𝜓) ↔ ∀𝑥𝐴 ¬ (𝜑𝜓))
4 df-an 383 . . . 4 ((∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 ¬ 𝜓) ↔ ¬ (∀𝑥𝐴 𝜑 → ¬ ∀𝑥𝐴 ¬ 𝜓))
51, 3, 43bitr3i 290 . . 3 (∀𝑥𝐴 ¬ (𝜑𝜓) ↔ ¬ (∀𝑥𝐴 𝜑 → ¬ ∀𝑥𝐴 ¬ 𝜓))
65con2bii 346 . 2 ((∀𝑥𝐴 𝜑 → ¬ ∀𝑥𝐴 ¬ 𝜓) ↔ ¬ ∀𝑥𝐴 ¬ (𝜑𝜓))
7 dfrex2 3143 . . 3 (∃𝑥𝐴 𝜓 ↔ ¬ ∀𝑥𝐴 ¬ 𝜓)
87imbi2i 325 . 2 ((∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓) ↔ (∀𝑥𝐴 𝜑 → ¬ ∀𝑥𝐴 ¬ 𝜓))
9 dfrex2 3143 . 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 382  wral 3060  wrex 3061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884
This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1852  df-ral 3065  df-rex 3066
This theorem is referenced by:  r19.36v  3232  r19.37  3233  r19.43  3240  r19.37zv  4206  r19.36zv  4211  iinexg  4952  bndndx  11492  nmobndseqi  27968  nmobndseqiALT  27969  r19.36vf  39839
  Copyright terms: Public domain W3C validator