Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  r19.37 Structured version   Visualization version   GIF version

Theorem r19.37 3272
 Description: Restricted quantifier version of one direction of 19.37 2218. (The other direction does not hold when 𝐴 is empty.) (Contributed by FL, 13-May-2012.) (Revised by Mario Carneiro, 11-Dec-2016.)
Hypothesis
Ref Expression
r19.37.1 𝑥𝜑
Assertion
Ref Expression
r19.37 (∃𝑥𝐴 (𝜑𝜓) → (𝜑 → ∃𝑥𝐴 𝜓))

Proof of Theorem r19.37
StepHypRef Expression
1 r19.35 3270 . 2 (∃𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))
2 r19.37.1 . . . 4 𝑥𝜑
3 ax-1 6 . . . 4 (𝜑 → (𝑥𝐴𝜑))
42, 3ralrimi 3139 . . 3 (𝜑 → ∀𝑥𝐴 𝜑)
54imim1i 63 . 2 ((∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓) → (𝜑 → ∃𝑥𝐴 𝜓))
61, 5sylbi 209 1 (∃𝑥𝐴 (𝜑𝜓) → (𝜑 → ∃𝑥𝐴 𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  Ⅎwnf 1827   ∈ wcel 2107  ∀wral 3090  ∃wrex 3091 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-12 2163 This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1824  df-nf 1828  df-ral 3095  df-rex 3096 This theorem is referenced by:  r19.37v  3273  ss2iundf  38922
 Copyright terms: Public domain W3C validator