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Theorem rmobida 2496
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 425 . . 3 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
41, 3mobid 1935 . 2 (𝜑 → (∃*𝑥(𝑥𝐴𝜓) ↔ ∃*𝑥(𝑥𝐴𝜒)))
5 df-rmo 2314 . 2 (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥(𝑥𝐴𝜓))
6 df-rmo 2314 . 2 (∃*𝑥𝐴 𝜒 ↔ ∃*𝑥(𝑥𝐴𝜒))
74, 5, 63bitr4g 212 1 (𝜑 → (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥𝐴 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98  wnf 1349  wcel 1393  ∃*wmo 1901  ∃*wrmo 2309
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-eu 1903  df-mo 1904  df-rmo 2314
This theorem is referenced by:  rmobidva  2497
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