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Theorem rmobidva 2651
Description: Formula-building rule for restricted existential quantifier (deduction form). (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 1515 . 2 𝑥𝜑
2 rmobidva.1 . 2 ((𝜑𝑥𝐴) → (𝜓𝜒))
31, 2rmobida 2650 1 (𝜑 → (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥𝐴 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wcel 2135  ∃*wrmo 2445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1434  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-4 1497  ax-17 1513  ax-ial 1521
This theorem depends on definitions:  df-bi 116  df-nf 1448  df-eu 2016  df-mo 2017  df-rmo 2450
This theorem is referenced by:  rmobidv  2652
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