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 3109
Description: Restricted quantifier version of 19.35 1881. (Contributed by NM, 20-Sep-2003.) (Proof shortened by Wolf Lammen, 22-Dec-2024.)
Assertion
Ref Expression
r19.35 (∃𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))

Proof of Theorem r19.35
StepHypRef Expression
1 pm5.5 362 . . . 4 (𝜑 → ((𝜑𝜓) ↔ 𝜓))
21ralrexbid 3107 . . 3 (∀𝑥𝐴 𝜑 → (∃𝑥𝐴 (𝜑𝜓) ↔ ∃𝑥𝐴 𝜓))
32biimpcd 248 . 2 (∃𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))
4 rexnal 3101 . . . 4 (∃𝑥𝐴 ¬ 𝜑 ↔ ¬ ∀𝑥𝐴 𝜑)
5 pm2.21 123 . . . . 5 𝜑 → (𝜑𝜓))
65reximi 3085 . . . 4 (∃𝑥𝐴 ¬ 𝜑 → ∃𝑥𝐴 (𝜑𝜓))
74, 6sylbir 234 . . 3 (¬ ∀𝑥𝐴 𝜑 → ∃𝑥𝐴 (𝜑𝜓))
8 ax-1 6 . . . 4 (𝜓 → (𝜑𝜓))
98reximi 3085 . . 3 (∃𝑥𝐴 𝜓 → ∃𝑥𝐴 (𝜑𝜓))
107, 9ja 186 . 2 ((∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓) → ∃𝑥𝐴 (𝜑𝜓))
113, 10impbii 208 1 (∃𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wral 3062  wrex 3071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-ral 3063  df-rex 3072
This theorem is referenced by:  r19.43  3123  r19.37v  3182  r19.36v  3184  r19.37  3260  r19.37zv  4502  r19.36zv  4507  iinexg  5342  bndndx  12471  nmobndseqi  30032  nmobndseqiALT  30033  unielss  41967  r19.36vf  43825
  Copyright terms: Public domain W3C validator