Theorem rmobidv 3320

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

Hypothesis

Ref	Expression
rmobidv.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

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

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)

Proof of Theorem rmobidv

Step	Hyp	Ref	Expression
1		rmobidv.1	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2	1	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))
3	2	rmobidva 3319	1 ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ↔ wb 205   ∈ wcel 2108  ∃*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  ax-5 1914  ax-6 1972  ax-7 2012  ax-12 2173

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-nf 1788  df-mo 2540  df-rmo 3071

This theorem is referenced by:  rmoeqd  3342  brdom7disj  10218  ddemeas  32104  poimirlem26  35730