Theorem rexrot4 2520

Description: Rotate existential restricted 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,𝑦,𝐶   𝑥,𝑧,𝐷,𝑦

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 2519	. . 3 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐷 𝜑 ↔ ∃𝑤 ∈ 𝐷 ∃𝑧 ∈ 𝐶 ∃𝑦 ∈ 𝐵 𝜑)
2	1	rexbii 2373	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐷 𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑤 ∈ 𝐷 ∃𝑧 ∈ 𝐶 ∃𝑦 ∈ 𝐵 𝜑)
3		rexcom13 2519	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑤 ∈ 𝐷 ∃𝑧 ∈ 𝐶 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐷 ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)
4	2, 3	bitri 182	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐷 𝜑 ↔ ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐷 ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)

Syntax hints:   ↔ wb 103  ∃wrex 2349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354

This theorem is referenced by: (None)