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Metamath Proof Explorer

Theorem 19.42vv

Description: Version of 19.42 with two quantifiers and a disjoint variable condition requiring fewer axioms. (Contributed by NM, 16-Mar-1995)

Ref Expression
Assertion 19.42vv ( ∃ 𝑥 ∃ 𝑦 ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜑 ∧ ∃ 𝑥 ∃ 𝑦 𝜓 ) )

Proof

Step Hyp Ref Expression
1 exdistr ⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝜑 ∧ 𝜓 ) ↔ ∃ 𝑥 ( 𝜑 ∧ ∃ 𝑦 𝜓 ) )
2 19.42v ⊢ ( ∃ 𝑥 ( 𝜑 ∧ ∃ 𝑦 𝜓 ) ↔ ( 𝜑 ∧ ∃ 𝑥 ∃ 𝑦 𝜓 ) )
3 1 2 bitri ⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜑 ∧ ∃ 𝑥 ∃ 𝑦 𝜓 ) )