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Theorem rmobida 3304
Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 16-Jun-2017.)
Hypotheses
Ref Expression
rmobida.1 𝑥𝜑
rmobida.2 ((𝜑𝑥𝐴) → (𝜓𝜒))
Assertion
Ref Expression
rmobida (𝜑 → (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥𝐴 𝜒))

Proof of Theorem rmobida
StepHypRef Expression
1 rmobida.1 . . 3 𝑥𝜑
2 rmobida.2 . . . 4 ((𝜑𝑥𝐴) → (𝜓𝜒))
32pm5.32da 582 . . 3 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
41, 3mobid 2549 . 2 (𝜑 → (∃*𝑥(𝑥𝐴𝜓) ↔ ∃*𝑥(𝑥𝐴𝜒)))
5 df-rmo 3069 . 2 (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥(𝑥𝐴𝜓))
6 df-rmo 3069 . 2 (∃*𝑥𝐴 𝜒 ↔ ∃*𝑥(𝑥𝐴𝜒))
74, 5, 63bitr4g 317 1 (𝜑 → (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wnf 1791  wcel 2110  ∃*wmo 2537  ∃*wrmo 3064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-12 2175
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-nf 1792  df-mo 2539  df-rmo 3069
This theorem is referenced by:  rmobidva  3305  reuan  3808
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