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Metamath Proof Explorer

Theorem 2rexbidv

Description: Formula-building rule for restricted existential quantifiers (deduction form). (Contributed by NM, 28-Jan-2006)

Ref Expression
Hypothesis 2ralbidv.1 ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
Assertion 2rexbidv ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜓 ↔ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜒 ) )

Proof

Step Hyp Ref Expression
1 2ralbidv.1 ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2 1 rexbidv ⊢ ( 𝜑 → ( ∃ 𝑦 ∈ 𝐵 𝜓 ↔ ∃ 𝑦 ∈ 𝐵 𝜒 ) )
3 2 rexbidv ⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜓 ↔ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜒 ) )