MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  r19.27z Structured version   Visualization version   GIF version

Theorem r19.27z 4453
Description: Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 26-Oct-2010.)
Hypothesis
Ref Expression
r19.27z.1 𝑥𝜓
Assertion
Ref Expression
r19.27z (𝐴 ≠ ∅ → (∀𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑𝜓)))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem r19.27z
StepHypRef Expression
1 r19.27z.1 . . . 4 𝑥𝜓
21r19.3rz 4445 . . 3 (𝐴 ≠ ∅ → (𝜓 ↔ ∀𝑥𝐴 𝜓))
32anbi2d 628 . 2 (𝐴 ≠ ∅ → ((∀𝑥𝐴 𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓)))
4 r19.26 3175 . 2 (∀𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓))
53, 4syl6rbbr 291 1 (𝐴 ≠ ∅ → (∀𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wnf 1777  wne 3021  wral 3143  c0 4295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-dif 3943  df-nul 4296
This theorem is referenced by:  r19.27zv  4454  raaan  4463  raaan2  4467
  Copyright terms: Public domain W3C validator