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

Theorem reubiia 3381
Description: Formula-building rule for restricted existential quantifier (inference form). (Contributed by NM, 14-Nov-2004.)
Hypothesis
Ref Expression
rmobiia.1 (𝑥𝐴 → (𝜑𝜓))
Assertion
Ref Expression
reubiia (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐴 𝜓)

Proof of Theorem reubiia
StepHypRef Expression
1 rmobiia.1 . . . 4 (𝑥𝐴 → (𝜑𝜓))
21pm5.32i 573 . . 3 ((𝑥𝐴𝜑) ↔ (𝑥𝐴𝜓))
32eubii 2577 . 2 (∃!𝑥(𝑥𝐴𝜑) ↔ ∃!𝑥(𝑥𝐴𝜓))
4 df-reu 3375 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
5 df-reu 3375 . 2 (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥(𝑥𝐴𝜓))
63, 4, 53bitr4i 302 1 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wcel 2104  ∃!weu 2560  ∃!wreu 3372
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809
This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1780  df-mo 2532  df-eu 2561  df-reu 3375
This theorem is referenced by:  reubii  3383  reuanid  3385  riotaxfrd  7402  opreuopreu  8022  infempty  9504
  Copyright terms: Public domain W3C validator