Theorem bireu 1320

Description: Formula-building rule for restricted existential quantifier (inference rule).

Hypothesis

Ref	Expression
bireu.1	⊢ (φ ↔ ψ)

Assertion

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

Proof of Theorem bireu

Step	Hyp	Ref	Expression
1		bireu.1	. . 3 ⊢ (φ ↔ ψ)
2	1	a1i 7	. 2 ⊢ (x ∈ A → (φ ↔ ψ))
3	2	bireua 1319	1 ⊢ (∃!x ∈ A φ ↔ ∃!x ∈ A ψ)

Syntax hints:   ↔ wb 127   ∈ wcel 1092  ∃!wreu 1203

This theorem is referenced by:  aceq2 3554  uzwo3 4616

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677  ax-9 799  ax-12 802  ax-17 925

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-eu 1009  df-reu 1207

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