Theorem rexbidv2

Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 22-May-1999)

	Ref	Expression
Hypothesis	rexbidv2.1	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝜒 ) ) )
Assertion	rexbidv2	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 ↔ ∃ 𝑥 ∈ 𝐵 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		rexbidv2.1	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝜒 ) ) )
2	1	exbidv	⊢ ( 𝜑 → ( ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜒 ) ) )
3		df-rex	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜓 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) )
4		df-rex	⊢ ( ∃ 𝑥 ∈ 𝐵 𝜒 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜒 ) )
5	2 3 4	3bitr4g	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 ↔ ∃ 𝑥 ∈ 𝐵 𝜒 ) )