Theorem rexbida

Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 6-Oct-2003)

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

Proof

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