Theorem rexrot4 2718

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 2717	. . 3 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐷 𝜑 ↔ ∃𝑤 ∈ 𝐷 ∃𝑧 ∈ 𝐶 ∃𝑦 ∈ 𝐵 𝜑)
2	1	rexbii 2557	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐷 𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑤 ∈ 𝐷 ∃𝑧 ∈ 𝐶 ∃𝑦 ∈ 𝐵 𝜑)
3		rexcom13 2717	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑤 ∈ 𝐷 ∃𝑧 ∈ 𝐶 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐷 ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)
4	2, 3	bitri 184	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐷 𝜑 ↔ ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐷 ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)

Syntax hints:   ↔ wb 105  ∃wrex 2529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534

This theorem is referenced by: (None)