Theorem rmobii 3374

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 3372	1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐴 𝜓)

This proof depends on syntax axioms:   ↔ wb 209   ∈ wcel 2145  ∃*wrmo 3365

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-mo 2565  df-rmo 3366

This theorem is used by:  2reu5a  3702  reuxfrd  3706  brdom7disj  10603  2sqreulem4  27774  nomaxmo  28048  reuxfrdf  33080  cvmlift2lem13  36059  ineccnvmo  39269  dfeldisj5  39725  lubeldm2  50033  glbeldm2  50034