MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rexcom4a Structured version   Visualization version   GIF version

Theorem rexcom4a 3198
Description: Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)
Assertion
Ref Expression
rexcom4a (∃𝑥𝑦𝐴 (𝜑𝜓) ↔ ∃𝑦𝐴 (𝜑 ∧ ∃𝑥𝜓))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑦   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑥,𝑦)   𝐴(𝑦)

Proof of Theorem rexcom4a
StepHypRef Expression
1 rexcom4 3197 . 2 (∃𝑦𝐴𝑥(𝜑𝜓) ↔ ∃𝑥𝑦𝐴 (𝜑𝜓))
2 19.42v 1912 . . 3 (∃𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))
32rexbii 3195 . 2 (∃𝑦𝐴𝑥(𝜑𝜓) ↔ ∃𝑦𝐴 (𝜑 ∧ ∃𝑥𝜓))
41, 3bitr3i 269 1 (∃𝑥𝑦𝐴 (𝜑𝜓) ↔ ∃𝑦𝐴 (𝜑 ∧ ∃𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 387  wex 1742  wrex 3090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-11 2093
This theorem depends on definitions:  df-bi 199  df-an 388  df-ex 1743  df-rex 3095
This theorem is referenced by:  rexcom4b  3446  bj-rexcom4bv  33688  bj-rexcom4b  33689
  Copyright terms: Public domain W3C validator