Theorem reubiia 1784

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 647	. . 3 ⊢ ((x ∈ A ⋀ φ) ↔ (x ∈ A ⋀ ψ))
3	2	eubii 1389	. 2 ⊢ (∃!x(x ∈ A ⋀ φ) ↔ ∃!x(x ∈ A ⋀ ψ))
4		df-reu 1654	. 2 ⊢ (∃!x ∈ A φ ↔ ∃!x(x ∈ A ⋀ φ))
5		df-reu 1654	. 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 960  ∃!weu 1382  ∃!wreu 1650

This theorem is referenced by:  reubii 1785  reuxfr2 2909  reuxfr 2910  reuunixfr 2912  pjtheu2 9245

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 965  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-eu 1384  df-reu 1654

Copyright terms: Public domain