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Theorem rmobidv 3108
 Description: Formula-building rule for restricted existential quantifier (deduction rule). (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 480 . 2 ((𝜑𝑥𝐴) → (𝜓𝜒))
32rmobidva 3107 1 (𝜑 → (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥𝐴 𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∈ wcel 1977  ∃*wrmo 2899 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-nf 1701  df-eu 2462  df-mo 2463  df-rmo 2904 This theorem is referenced by:  rmoeqd  3126  brdom7disj  9234  ddemeas  29626  poimirlem26  32605
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