Theorem reubiia 2488

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 427	. . 3 ⊢ ((x ∈ A ∧ φ) ↔ (x ∈ A ∧ ψ))
3	2	eubii 1906	. 2 ⊢ (∃!x(x ∈ A ∧ φ) ↔ ∃!x(x ∈ A ∧ ψ))
4		df-reu 2307	. 2 ⊢ (∃!x ∈ A φ ↔ ∃!x(x ∈ A ∧ φ))
5		df-reu 2307	. 2 ⊢ (∃!x ∈ A ψ ↔ ∃!x(x ∈ A ∧ ψ))
6	3, 4, 5	3bitr4i 201	1 ⊢ (∃!x ∈ A φ ↔ ∃!x ∈ A ψ)

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∈ wcel 1390  ∃!weu 1897  ∃!wreu 2302

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-eu 1900  df-reu 2307

This theorem is referenced by:  reubii  2489