MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  r19.41 Structured version   Visualization version   GIF version

Theorem r19.41 3271
Description: Restricted quantifier version of 19.41 2270. See r19.41v 3270 for a version with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 1-Nov-2010.)
Hypothesis
Ref Expression
r19.41.1 𝑥𝜓
Assertion
Ref Expression
r19.41 (∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓))

Proof of Theorem r19.41
StepHypRef Expression
1 anass 461 . . . 4 (((𝑥𝐴𝜑) ∧ 𝜓) ↔ (𝑥𝐴 ∧ (𝜑𝜓)))
21exbii 1944 . . 3 (∃𝑥((𝑥𝐴𝜑) ∧ 𝜓) ↔ ∃𝑥(𝑥𝐴 ∧ (𝜑𝜓)))
3 r19.41.1 . . . 4 𝑥𝜓
4319.41 2270 . . 3 (∃𝑥((𝑥𝐴𝜑) ∧ 𝜓) ↔ (∃𝑥(𝑥𝐴𝜑) ∧ 𝜓))
52, 4bitr3i 269 . 2 (∃𝑥(𝑥𝐴 ∧ (𝜑𝜓)) ↔ (∃𝑥(𝑥𝐴𝜑) ∧ 𝜓))
6 df-rex 3095 . 2 (∃𝑥𝐴 (𝜑𝜓) ↔ ∃𝑥(𝑥𝐴 ∧ (𝜑𝜓)))
7 df-rex 3095 . . 3 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
87anbi1i 618 . 2 ((∃𝑥𝐴 𝜑𝜓) ↔ (∃𝑥(𝑥𝐴𝜑) ∧ 𝜓))
95, 6, 83bitr4i 295 1 (∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 385  wex 1875  wnf 1879  wcel 2157  wrex 3090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-12 2213
This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876  df-nf 1880  df-rex 3095
This theorem is referenced by:  iunin1f  29892
  Copyright terms: Public domain W3C validator