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Metamath Proof Explorer

Theorem 3exbii

Description: Inference adding three existential quantifiers to both sides of an equivalence. (Contributed by NM, 2-May-1995)

Ref Expression
Hypothesis 3exbii.1 ⊢ ( 𝜑 ↔ 𝜓 )
Assertion 3exbii ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 𝜓 )

Proof

Step Hyp Ref Expression
1 3exbii.1 ⊢ ( 𝜑 ↔ 𝜓 )
2 1 exbii ⊢ ( ∃ 𝑧 𝜑 ↔ ∃ 𝑧 𝜓 )
3 2 2exbii ⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 𝜓 )