Theorem reubii 2798

Description: Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 22-Oct-1999.)

Hypothesis

Ref	Expression
reubii.1	⊢ (φ ↔ ψ)

Assertion

Ref	Expression
reubii	⊢ (∃!x ∈ A φ ↔ ∃!x ∈ A ψ)

Proof of Theorem reubii

Step	Hyp	Ref	Expression
1		reubii.1	. . 3 ⊢ (φ ↔ ψ)
2	1	a1i 10	. 2 ⊢ (x ∈ A → (φ ↔ ψ))
3	2	reubiia 2797	1 ⊢ (∃!x ∈ A φ ↔ ∃!x ∈ A ψ)

Syntax hints:   ↔ wb 176   ∈ wcel 1710  ∃!wreu 2617

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-tru 1319  df-ex 1542  df-nf 1545  df-eu 2208  df-reu 2622

This theorem is referenced by:  2reu5lem1  3042