Theorem reubida

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

	Ref	Expression
Hypotheses	reubida.1	⊢ Ⅎ 𝑥 𝜑
	reubida.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
Assertion	reubida	⊢ ( 𝜑 → ( ∃! 𝑥 ∈ 𝐴 𝜓 ↔ ∃! 𝑥 ∈ 𝐴 𝜒 ) )

Proof

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