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Theorem rmobiia 2516
Description: Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 16-Jun-2017.)
Hypothesis
Ref Expression
rmobiia.1 (𝑥𝐴 → (𝜑𝜓))
Assertion
Ref Expression
rmobiia (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐴 𝜓)

Proof of Theorem rmobiia
StepHypRef Expression
1 rmobiia.1 . . . 4 (𝑥𝐴 → (𝜑𝜓))
21pm5.32i 435 . . 3 ((𝑥𝐴𝜑) ↔ (𝑥𝐴𝜓))
32mobii 1953 . 2 (∃*𝑥(𝑥𝐴𝜑) ↔ ∃*𝑥(𝑥𝐴𝜓))
4 df-rmo 2331 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
5 df-rmo 2331 . 2 (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥(𝑥𝐴𝜓))
63, 4, 53bitr4i 205 1 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐴 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wcel 1409  ∃*wmo 1917  ∃*wrmo 2326
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-eu 1919  df-mo 1920  df-rmo 2331
This theorem is referenced by:  rmobii  2517
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