Theorem rmobii 3384

Description: Formula-building rule for restricted at-most-one quantifier (inference form). (Contributed by NM, 16-Jun-2017.)

Hypothesis

Ref	Expression
rmobii.1	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
rmobii	⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐴 𝜓)

Proof of Theorem rmobii

Step	Hyp	Ref	Expression
1		rmobii.1	. . 3 ⊢ (𝜑 ↔ 𝜓)
2	1	a1i 11	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓))
3	2	rmobiia 3382	1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐴 𝜓)

Syntax hints:   ↔ wb 209   ∈ wcel 2149  ∃*wrmo 3375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-mo 2573  df-rmo 3376

This theorem is referenced by:  2reu5a  3716  reuxfrd  3720  brdom7disj  10511  2sqreulem4  27580  nomaxmo  27824  reuxfrdf  32774  cvmlift2lem13  35702  ineccnvmo  38891  dfeldisj5  39347  lubeldm2  49614  glbeldm2  49615