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Theorem rmobidva 2542
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 1462 . 2 𝑥𝜑
2 rmobidva.1 . 2 ((𝜑𝑥𝐴) → (𝜓𝜒))
31, 2rmobida 2541 1 (𝜑 → (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥𝐴 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wcel 1434  ∃*wrmo 2352
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 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-eu 1945  df-mo 1946  df-rmo 2357
This theorem is referenced by:  rmobidv  2543
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