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Theorem rmobidv 3296
Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 16-Jun-2017.)
Hypothesis
Ref Expression
rmobidv.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
rmobidv (𝜑 → (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥𝐴 𝜒))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)

Proof of Theorem rmobidv
StepHypRef Expression
1 rmobidv.1 . . 3 (𝜑 → (𝜓𝜒))
21adantr 484 . 2 ((𝜑𝑥𝐴) → (𝜓𝜒))
32rmobidva 3295 1 (𝜑 → (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wcel 2113  ∃*wrmo 3056
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-12 2178
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1787  df-nf 1791  df-mo 2540  df-rmo 3061
This theorem is referenced by:  rmoeqd  3318  brdom7disj  10024  ddemeas  31766  poimirlem26  35415
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