Theorem reubiia 2797

Description: Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 14-Nov-2004.)

Hypothesis

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

Assertion

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

Proof of Theorem reubiia

Step	Hyp	Ref	Expression
1		reubiia.1	. . . 4 ⊢ (x ∈ A → (φ ↔ ψ))
2	1	pm5.32i 618	. . 3 ⊢ ((x ∈ A ∧ φ) ↔ (x ∈ A ∧ ψ))
3	2	eubii 2213	. 2 ⊢ (∃!x(x ∈ A ∧ φ) ↔ ∃!x(x ∈ A ∧ ψ))
4		df-reu 2622	. 2 ⊢ (∃!x ∈ A φ ↔ ∃!x(x ∈ A ∧ φ))
5		df-reu 2622	. 2 ⊢ (∃!x ∈ A ψ ↔ ∃!x(x ∈ A ∧ ψ))
6	3, 4, 5	3bitr4i 268	1 ⊢ (∃!x ∈ A φ ↔ ∃!x ∈ A ψ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∈ wcel 1710  ∃!weu 2204  ∃!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:  reubii  2798