Theorem rmobii 3385

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 3383	1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐴 𝜓)

Syntax hints:   ↔ wb 205   ∈ wcel 2107  ∃*wrmo 3376

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812

This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-mo 2535  df-rmo 3377

This theorem is referenced by:  2reu5a  3741  reuxfrd  3745  brdom7disj  10526  2sqreulem4  26957  nomaxmo  27201  reuxfrdf  31762  cvmlift2lem13  34337  ineccnvmo  37274  dfeldisj5  37639  lubeldm2  47637  glbeldm2  47638