Theorem reubii 1829

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

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 8	. 2 ⊢ (x ∈ A → (φ ↔ ψ))
3	2	reubiia 1828	1 ⊢ (∃!x ∈ A φ ↔ ∃!x ∈ A ψ)

Syntax hints:   ↔ wb 153   ∈ wcel 999  ∃!wreu 1694

This theorem is referenced by:  aceq2 4793  infmsup 6150  uzwo3 6303  cnlnadjlem3 10085  cnlnadjlem4 10086  cnlnadjlem5 10087

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1004  ax-12 1009  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164

This theorem depends on definitions:  df-bi 154  df-an 232  df-ex 1022  df-eu 1424  df-reu 1698

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