Theorem reubida 2485

Description: Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by Mario Carneiro, 19-Nov-2016.)

Hypotheses

Ref	Expression
reubida.1	⊢ Ⅎxφ
reubida.2	⊢ ((φ ∧ x ∈ A) → (ψ ↔ χ))

Assertion

Ref	Expression
reubida	⊢ (φ → (∃!x ∈ A ψ ↔ ∃!x ∈ A χ))

Proof of Theorem reubida

Step	Hyp	Ref	Expression
1		reubida.1	. . 3 ⊢ Ⅎxφ
2		reubida.2	. . . 4 ⊢ ((φ ∧ x ∈ A) → (ψ ↔ χ))
3	2	pm5.32da 425	. . 3 ⊢ (φ → ((x ∈ A ∧ ψ) ↔ (x ∈ A ∧ χ)))
4	1, 3	eubid 1904	. 2 ⊢ (φ → (∃!x(x ∈ A ∧ ψ) ↔ ∃!x(x ∈ A ∧ χ)))
5		df-reu 2307	. 2 ⊢ (∃!x ∈ A ψ ↔ ∃!x(x ∈ A ∧ ψ))
6		df-reu 2307	. 2 ⊢ (∃!x ∈ A χ ↔ ∃!x(x ∈ A ∧ χ))
7	4, 5, 6	3bitr4g 212	1 ⊢ (φ → (∃!x ∈ A ψ ↔ ∃!x ∈ A χ))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  Ⅎwnf 1346   ∈ 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-nf 1347  df-eu 1900  df-reu 2307

This theorem is referenced by:  reubidva  2486