Theorem rmobidva 2800

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

Hypothesis

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

Assertion

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

Distinct variable group:   φ,x

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

Proof of Theorem rmobidva

Step	Hyp	Ref	Expression
1		nfv 1619	. 2 ⊢ Ⅎxφ
2		rmobidva.1	. 2 ⊢ ((φ ∧ x ∈ A) → (ψ ↔ χ))
3	1, 2	rmobida 2799	1 ⊢ (φ → (∃*x ∈ A ψ ↔ ∃*x ∈ A χ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∈ wcel 1710  ∃*wrmo 2618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-eu 2208  df-mo 2209  df-rmo 2623

This theorem is referenced by:  rmobidv  2801