Theorem bireua 1319

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

Hypothesis

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

Assertion

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

Proof of Theorem bireua

Step	Hyp	Ref	Expression
1		bireua.1	. . . 4 ⊢ (x ∈ A → (φ ↔ ψ))
2	1	pm5.32i 489	. . 3 ⊢ ((x ∈ A ∧ φ) ↔ (x ∈ A ∧ ψ))
3	2	bieu 1014	. 2 ⊢ (∃!x(x ∈ A ∧ φ) ↔ ∃!x(x ∈ A ∧ ψ))
4		df-reu 1207	. 2 ⊢ (∃!x ∈ A φ ↔ ∃!x(x ∈ A ∧ φ))
5		df-reu 1207	. 2 ⊢ (∃!x ∈ A ψ ↔ ∃!x(x ∈ A ∧ ψ))
6	3, 4, 5	3bitr4 158	1 ⊢ (∃!x ∈ A φ ↔ ∃!x ∈ A ψ)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃!weu 1007   ∈ wcel 1092  ∃!wreu 1203

This theorem is referenced by:  bireu 1320  reuxfr2 1579  reuxfr 1580

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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