Theorem rmobii 3353

Description: Formula-building rule for restricted existential 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 3352	1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐴 𝜓)

Syntax hints:   ↔ wb 207   ∈ wcel 2079  ∃*wrmo 3106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1760  df-mo 2574  df-rmo 3111

This theorem is referenced by:  2reu5a  3664  reuxfrd  3668  brdom7disj  9788  2sqreulem4  25700  reuxfrdf  29934  cvmlift2lem13  32126  nomaxmo  32755  ineccnvmo  35093  dfeldisj5  35435