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Theorem rmobidv 2665
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 276 . 2 ((𝜑𝑥𝐴) → (𝜓𝜒))
32rmobidva 2664 1 (𝜑 → (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥𝐴 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wcel 2148  ∃*wrmo 2458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-eu 2029  df-mo 2030  df-rmo 2463
This theorem is referenced by:  rmoeqd  2683  disjxp1  6234
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