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Theorem reubiia 3157
Description: Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 14-Nov-2004.)
Hypothesis
Ref Expression
reubiia.1 (𝑥𝐴 → (𝜑𝜓))
Assertion
Ref Expression
reubiia (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐴 𝜓)

Proof of Theorem reubiia
StepHypRef Expression
1 reubiia.1 . . . 4 (𝑥𝐴 → (𝜑𝜓))
21pm5.32i 670 . . 3 ((𝑥𝐴𝜑) ↔ (𝑥𝐴𝜓))
32eubii 2520 . 2 (∃!𝑥(𝑥𝐴𝜑) ↔ ∃!𝑥(𝑥𝐴𝜓))
4 df-reu 2948 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
5 df-reu 2948 . 2 (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥(𝑥𝐴𝜓))
63, 4, 53bitr4i 292 1 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wcel 2030  ∃!weu 2498  ∃!wreu 2943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-12 2087
This theorem depends on definitions:  df-bi 197  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-eu 2502  df-reu 2948
This theorem is referenced by:  reubii  3158  riotaxfrd  6682  infempty  8453
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