Theorem rmobida 3106

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

Hypotheses

Ref	Expression
rmobida.1	⊢ Ⅎ𝑥𝜑
rmobida.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
rmobida	⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐴 𝜒))

Proof of Theorem rmobida

Step	Hyp	Ref	Expression
1		rmobida.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		rmobida.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))
3	2	pm5.32da 671	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)))
4	1, 3	mobid 2477	. 2 ⊢ (𝜑 → (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜒)))
5		df-rmo 2904	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
6		df-rmo 2904	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜒 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜒))
7	4, 5, 6	3bitr4g 302	1 ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  Ⅎwnf 1699   ∈ wcel 1977  ∃*wmo 2459  ∃*wrmo 2899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-nf 1701  df-eu 2462  df-mo 2463  df-rmo 2904

This theorem is referenced by:  rmobidva  3107  reuan  39829