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Theorem rmobiia 3298
Description: Formula-building rule for restricted existential quantifier (inference form). (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 578 . . 3 ((𝑥𝐴𝜑) ↔ (𝑥𝐴𝜓))
32mobii 2548 . 2 (∃*𝑥(𝑥𝐴𝜑) ↔ ∃*𝑥(𝑥𝐴𝜓))
4 df-rmo 3061 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
5 df-rmo 3061 . 2 (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥(𝑥𝐴𝜓))
63, 4, 53bitr4i 306 1 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wcel 2114  ∃*wmo 2538  ∃*wrmo 3056
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1787  df-mo 2540  df-rmo 3061
This theorem is referenced by:  rmobii  3299
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