Theorem rexrot4 3289

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 3287	. . 3 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐷 𝜑 ↔ ∃𝑤 ∈ 𝐷 ∃𝑧 ∈ 𝐶 ∃𝑦 ∈ 𝐵 𝜑)
2	1	rexbii 3179	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐷 𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑤 ∈ 𝐷 ∃𝑧 ∈ 𝐶 ∃𝑦 ∈ 𝐵 𝜑)
3		rexcom13 3287	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑤 ∈ 𝐷 ∃𝑧 ∈ 𝐶 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐷 ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)
4	2, 3	bitri 274	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐷 𝜑 ↔ ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐷 ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)

Syntax hints:   ↔ wb 205  ∃wrex 3066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-11 2157

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1786  df-rex 3071

This theorem is referenced by:  lsmspsn  20327