Theorem 4exdistrv 1989

Description: Distribute two pairs of existential quantifiers (over disjoint variables) over a conjunction. For a version with fewer disjoint variable conditions but requiring more axioms, see ee4anv 2385. (Contributed by BJ, 5-Jan-2023.)

Assertion

Ref	Expression
4exdistrv	⊢ (∃𝑥∃𝑧∃𝑦∃𝑤(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤𝜓))

Distinct variable groups:   𝜑,𝑤   𝜑,𝑧   𝜓,𝑦   𝜓,𝑥   𝑦,𝑤   𝑦,𝑧   𝑥,𝑤   𝑥,𝑧

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝜓(𝑧, 𝑤)

Proof of Theorem 4exdistrv

Step	Hyp	Ref	Expression
1		exdistrv 1988	. . 3 ⊢ (∃𝑦∃𝑤(𝜑 ∧ 𝜓) ↔ (∃𝑦𝜑 ∧ ∃𝑤𝜓))
2	1	2exbii 1882	. 2 ⊢ (∃𝑥∃𝑧∃𝑦∃𝑤(𝜑 ∧ 𝜓) ↔ ∃𝑥∃𝑧(∃𝑦𝜑 ∧ ∃𝑤𝜓))
3		exdistrv 1988	. 2 ⊢ (∃𝑥∃𝑧(∃𝑦𝜑 ∧ ∃𝑤𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤𝜓))
4	2, 3	bitri 278	1 ⊢ (∃𝑥∃𝑧∃𝑦∃𝑤(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤𝜓))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813

This theorem is used by:  3oalem3  32087