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Theorem reubiia 3309
 Description: Formula-building rule for restricted existential quantifier (inference form). (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 578 . . 3 ((𝑥𝐴𝜑) ↔ (𝑥𝐴𝜓))
32eubii 2605 . 2 (∃!𝑥(𝑥𝐴𝜑) ↔ ∃!𝑥(𝑥𝐴𝜓))
4 df-reu 3078 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
5 df-reu 3078 . 2 (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥(𝑥𝐴𝜓))
63, 4, 53bitr4i 306 1 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐴 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∈ wcel 2112  ∃!weu 2588  ∃!wreu 3073 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1783  df-mo 2558  df-eu 2589  df-reu 3078 This theorem is referenced by:  reubii  3310  riotaxfrd  7149  opreuopreu  7745  infempty  9018
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