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Theorem rmobidva 2497
 Description: Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 16-Jun-2017.)
Hypothesis
Ref Expression
rmobidva.1 ((𝜑𝑥𝐴) → (𝜓𝜒))
Assertion
Ref Expression
rmobidva (𝜑 → (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥𝐴 𝜒))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)

Proof of Theorem rmobidva
StepHypRef Expression
1 nfv 1421 . 2 𝑥𝜑
2 rmobidva.1 . 2 ((𝜑𝑥𝐴) → (𝜓𝜒))
31, 2rmobida 2496 1 (𝜑 → (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥𝐴 𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∈ wcel 1393  ∃*wrmo 2309 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-eu 1903  df-mo 1904  df-rmo 2314 This theorem is referenced by:  rmobidv  2498
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