MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  2reu2rex Structured version   Visualization version   GIF version

Theorem 2reu2rex 3430
Description: Double restricted existential uniqueness, analogous to 2eu2ex 2721. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
Assertion
Ref Expression
2reu2rex (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 → ∃𝑥𝐴𝑦𝐵 𝜑)

Proof of Theorem 2reu2rex
StepHypRef Expression
1 reurex 3429 . 2 (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 → ∃𝑥𝐴 ∃!𝑦𝐵 𝜑)
2 reurex 3429 . . 3 (∃!𝑦𝐵 𝜑 → ∃𝑦𝐵 𝜑)
32reximi 3240 . 2 (∃𝑥𝐴 ∃!𝑦𝐵 𝜑 → ∃𝑥𝐴𝑦𝐵 𝜑)
41, 3syl 17 1 (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 → ∃𝑥𝐴𝑦𝐵 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wrex 3136  ∃!wreu 3137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-eu 2647  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143
This theorem is referenced by:  2reu1  3878  2sqreultblem  25951  2sqreunnltblem  25954
  Copyright terms: Public domain W3C validator