Theorem rexrot4 3302

Description: Rotate four restricted existential quantifiers twice. (Contributed by NM, 8-Apr-2015.)

Assertion

Ref	Expression
rexrot4	⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐷 𝜑 ↔ ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐷 ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)

Distinct variable groups:   𝑧,𝑤,𝐴   𝑤,𝐵,𝑧   𝑥,𝑤,𝐶   𝑦,𝑤,𝐶   𝑥,𝑧,𝐷   𝑦,𝑧,𝐷

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑧, 𝑤)   𝐴(𝑥, 𝑦)   𝐵(𝑥, 𝑦)   𝐶(𝑧)   𝐷(𝑤)

Proof of Theorem rexrot4

Step	Hyp	Ref	Expression
1		rexcom13 3301	. . 3 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐷 𝜑 ↔ ∃𝑤 ∈ 𝐷 ∃𝑧 ∈ 𝐶 ∃𝑦 ∈ 𝐵 𝜑)
2	1	rexbii 3115	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐷 𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑤 ∈ 𝐷 ∃𝑧 ∈ 𝐶 ∃𝑦 ∈ 𝐵 𝜑)
3		rexcom13 3301	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑤 ∈ 𝐷 ∃𝑧 ∈ 𝐶 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐷 ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)
4	2, 3	bitri 278	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐷 𝜑 ↔ ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐷 ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)

This proof depends on syntax axioms:   ↔ wb 209  ∃wrex 3092

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  ax-11 2195

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-ral 3083  df-rex 3093

This theorem is used by:  lsmspsn  21242