Theorem rmobiia 3321

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

Hypothesis

Ref	Expression
rmobiia.1	⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓))

Assertion

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

Proof of Theorem rmobiia

Step	Hyp	Ref	Expression
1		rmobiia.1	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓))
2	1	pm5.32i 574	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ 𝜓))
3	2	mobii 2548	. 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
4		df-rmo 3071	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
5		df-rmo 3071	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
6	3, 4, 5	3bitr4i 302	1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∈ wcel 2108  ∃*wmo 2538  ∃*wrmo 3066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-mo 2540  df-rmo 3071

This theorem is referenced by:  rmobii  3322