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

Theorem r19.35 3333
 Description: Restricted quantifier version of 19.35 1879. (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 pm2.27 42 . . . . 5 (𝜑 → ((𝜑𝜓) → 𝜓))
21ralimi 3155 . . . 4 (∀𝑥𝐴 𝜑 → ∀𝑥𝐴 ((𝜑𝜓) → 𝜓))
3 rexim 3236 . . . 4 (∀𝑥𝐴 ((𝜑𝜓) → 𝜓) → (∃𝑥𝐴 (𝜑𝜓) → ∃𝑥𝐴 𝜓))
42, 3syl 17 . . 3 (∀𝑥𝐴 𝜑 → (∃𝑥𝐴 (𝜑𝜓) → ∃𝑥𝐴 𝜓))
54com12 32 . 2 (∃𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))
6 rexnal 3233 . . . 4 (∃𝑥𝐴 ¬ 𝜑 ↔ ¬ ∀𝑥𝐴 𝜑)
7 pm2.21 123 . . . . 5 𝜑 → (𝜑𝜓))
87reximi 3238 . . . 4 (∃𝑥𝐴 ¬ 𝜑 → ∃𝑥𝐴 (𝜑𝜓))
96, 8sylbir 238 . . 3 (¬ ∀𝑥𝐴 𝜑 → ∃𝑥𝐴 (𝜑𝜓))
10 ax-1 6 . . . 4 (𝜓 → (𝜑𝜓))
1110reximi 3238 . . 3 (∃𝑥𝐴 𝜓 → ∃𝑥𝐴 (𝜑𝜓))
129, 11ja 189 . 2 ((∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓) → ∃𝑥𝐴 (𝜑𝜓))
135, 12impbii 212 1 (∃𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wral 3133  ∃wrex 3134 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-ral 3138  df-rex 3139 This theorem is referenced by:  r19.36v  3334  r19.37  3335  r19.37v  3336  r19.43  3343  r19.37zv  4429  r19.36zv  4434  iinexg  5230  bndndx  11889  nmobndseqi  28558  nmobndseqiALT  28559  r19.36vf  41632
 Copyright terms: Public domain W3C validator