Theorem 2ex2rexrot 3298

Description: Rotate two existential quantifiers and two restricted existential quantifiers. (Contributed by AV, 9-Nov-2023.)

Assertion

Ref	Expression
2ex2rexrot	⊢ (∃𝑥∃𝑦∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∃𝑥∃𝑦𝜑)

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

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

Proof of Theorem 2ex2rexrot

Step	Hyp	Ref	Expression
1		rexcom4 3290	. . 3 ⊢ (∃𝑤 ∈ 𝐵 ∃𝑥∃𝑦𝜑 ↔ ∃𝑥∃𝑤 ∈ 𝐵 ∃𝑦𝜑)
2	1	rexbii 3110	. 2 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∃𝑥∃𝑦𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑥∃𝑤 ∈ 𝐵 ∃𝑦𝜑)
3		rexcom4 3290	. 2 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑥∃𝑤 ∈ 𝐵 ∃𝑦𝜑 ↔ ∃𝑥∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∃𝑦𝜑)
4		rexcom4 3290	. . . . 5 ⊢ (∃𝑤 ∈ 𝐵 ∃𝑦𝜑 ↔ ∃𝑦∃𝑤 ∈ 𝐵 𝜑)
5	4	rexbii 3110	. . . 4 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∃𝑦𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑦∃𝑤 ∈ 𝐵 𝜑)
6		rexcom4 3290	. . . 4 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑦∃𝑤 ∈ 𝐵 𝜑 ↔ ∃𝑦∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜑)
7	5, 6	bitri 278	. . 3 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∃𝑦𝜑 ↔ ∃𝑦∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜑)
8	7	exbii 1881	. 2 ⊢ (∃𝑥∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∃𝑦𝜑 ↔ ∃𝑥∃𝑦∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜑)
9	2, 3, 8	3bitrri 301	1 ⊢ (∃𝑥∃𝑦∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∃𝑥∃𝑦𝜑)

This proof depends on syntax axioms:   ↔ wb 209  ∃wex 1812  ∃wrex 3087

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 2194

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-rex 3088

This theorem is used by:  satfv1  36097