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Metamath Proof Explorer

Theorem ee4anv

Description: Distribute two pairs of existential quantifiers over a conjunction. For a version requiring fewer axioms but with additional disjoint variable conditions, see 4exdistrv . (Contributed by NM, 31-Jul-1995) Remove disjoint variable conditions on y , z and x , w . (Revised by Eric Schmidt, 26-Oct-2025)

Ref Expression
Assertion ee4anv ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( 𝜑 ∧ 𝜓 ) ↔ ( ∃ 𝑥 ∃ 𝑦 𝜑 ∧ ∃ 𝑧 ∃ 𝑤 𝜓 ) )

Proof

Step Hyp Ref Expression
1 excom ⊢ ( ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( 𝜑 ∧ 𝜓 ) ↔ ∃ 𝑧 ∃ 𝑦 ∃ 𝑤 ( 𝜑 ∧ 𝜓 ) )
2 1 exbii ⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( 𝜑 ∧ 𝜓 ) ↔ ∃ 𝑥 ∃ 𝑧 ∃ 𝑦 ∃ 𝑤 ( 𝜑 ∧ 𝜓 ) )
3 eeanv ⊢ ( ∃ 𝑦 ∃ 𝑤 ( 𝜑 ∧ 𝜓 ) ↔ ( ∃ 𝑦 𝜑 ∧ ∃ 𝑤 𝜓 ) )
4 3 2exbii ⊢ ( ∃ 𝑥 ∃ 𝑧 ∃ 𝑦 ∃ 𝑤 ( 𝜑 ∧ 𝜓 ) ↔ ∃ 𝑥 ∃ 𝑧 ( ∃ 𝑦 𝜑 ∧ ∃ 𝑤 𝜓 ) )
5 nfv ⊢ Ⅎ 𝑧 𝜑
6 5 nfex ⊢ Ⅎ 𝑧 ∃ 𝑦 𝜑
7 nfv ⊢ Ⅎ 𝑥 𝜓
8 7 nfex ⊢ Ⅎ 𝑥 ∃ 𝑤 𝜓
9 6 8 eean ⊢ ( ∃ 𝑥 ∃ 𝑧 ( ∃ 𝑦 𝜑 ∧ ∃ 𝑤 𝜓 ) ↔ ( ∃ 𝑥 ∃ 𝑦 𝜑 ∧ ∃ 𝑧 ∃ 𝑤 𝜓 ) )
10 2 4 9 3bitri ⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( 𝜑 ∧ 𝜓 ) ↔ ( ∃ 𝑥 ∃ 𝑦 𝜑 ∧ ∃ 𝑧 ∃ 𝑤 𝜓 ) )