Theorem reubida 2794

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 622	. . 3 ⊢ (φ → ((x ∈ A ∧ ψ) ↔ (x ∈ A ∧ χ)))
4	1, 3	eubid 2211	. 2 ⊢ (φ → (∃!x(x ∈ A ∧ ψ) ↔ ∃!x(x ∈ A ∧ χ)))
5		df-reu 2622	. 2 ⊢ (∃!x ∈ A ψ ↔ ∃!x(x ∈ A ∧ ψ))
6		df-reu 2622	. 2 ⊢ (∃!x ∈ A χ ↔ ∃!x(x ∈ A ∧ χ))
7	4, 5, 6	3bitr4g 279	1 ⊢ (φ → (∃!x ∈ A ψ ↔ ∃!x ∈ A χ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  Ⅎwnf 1544   ∈ wcel 1710  ∃!weu 2204  ∃!wreu 2617

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-eu 2208  df-reu 2622

This theorem is referenced by:  reubidva  2795