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Theorem reubiia 2551
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 442 . . 3 ((𝑥𝐴𝜑) ↔ (𝑥𝐴𝜓))
32eubii 1957 . 2 (∃!𝑥(𝑥𝐴𝜑) ↔ ∃!𝑥(𝑥𝐴𝜓))
4 df-reu 2366 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
5 df-reu 2366 . 2 (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥(𝑥𝐴𝜓))
63, 4, 53bitr4i 210 1 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐴 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wcel 1438  ∃!weu 1948  ∃!wreu 2361
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-eu 1951  df-reu 2366
This theorem is referenced by:  reubii  2552
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