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Theorem rmobiia 2570
 Description: Formula-building rule for restricted existential quantifier (inference form). (Contributed by NM, 16-Jun-2017.)
Hypothesis
Ref Expression
rmobiia.1 (𝑥𝐴 → (𝜑𝜓))
Assertion
Ref Expression
rmobiia (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐴 𝜓)

Proof of Theorem rmobiia
StepHypRef Expression
1 rmobiia.1 . . . 4 (𝑥𝐴 → (𝜑𝜓))
21pm5.32i 443 . . 3 ((𝑥𝐴𝜑) ↔ (𝑥𝐴𝜓))
32mobii 1992 . 2 (∃*𝑥(𝑥𝐴𝜑) ↔ ∃*𝑥(𝑥𝐴𝜓))
4 df-rmo 2378 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
5 df-rmo 2378 . 2 (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥(𝑥𝐴𝜓))
63, 4, 53bitr4i 211 1 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐴 𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∈ wcel 1445  ∃*wmo 1956  ∃*wrmo 2373 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1388  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-4 1452  ax-17 1471  ax-ial 1479 This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-eu 1958  df-mo 1959  df-rmo 2378 This theorem is referenced by:  rmobii  2571
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