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Theorem rmobida 3159
Description: Formula-building rule for restricted existential quantifier (deduction rule). (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 674 . . 3 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
41, 3mobid 2517 . 2 (𝜑 → (∃*𝑥(𝑥𝐴𝜓) ↔ ∃*𝑥(𝑥𝐴𝜒)))
5 df-rmo 2949 . 2 (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥(𝑥𝐴𝜓))
6 df-rmo 2949 . 2 (∃*𝑥𝐴 𝜒 ↔ ∃*𝑥(𝑥𝐴𝜒))
74, 5, 63bitr4g 303 1 (𝜑 → (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wnf 1748  wcel 2030  ∃*wmo 2499  ∃*wrmo 2944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-12 2087
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-nf 1750  df-eu 2502  df-mo 2503  df-rmo 2949
This theorem is referenced by:  rmobidva  3160  reuan  41501
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