Theorem reubiia 1778

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

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 644	. . 3 ⊢ ((x ∈ A ⋀ φ) ↔ (x ∈ A ⋀ ψ))
3	2	eubii 1385	. 2 ⊢ (∃!x(x ∈ A ⋀ φ) ↔ ∃!x(x ∈ A ⋀ ψ))
4		df-reu 1648	. 2 ⊢ (∃!x ∈ A φ ↔ ∃!x(x ∈ A ⋀ φ))
5		df-reu 1648	. 2 ⊢ (∃!x ∈ A ψ ↔ ∃!x(x ∈ A ⋀ ψ))
6	3, 4, 5	3bitr4 183	1 ⊢ (∃!x ∈ A φ ↔ ∃!x ∈ A ψ)

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223   ∈ wcel 956  ∃!weu 1378  ∃!wreu 1644

This theorem is referenced by:  reubii 1779  reuxfr2 2898  reuxfr 2899  reuunixfr 2901  pjtheu2 9188

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 961  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-eu 1380  df-reu 1648

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