Theorem 2ex2rexrot 3250

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 3249	. . 3 ⊢ (∃𝑤 ∈ 𝐵 ∃𝑥∃𝑦𝜑 ↔ ∃𝑥∃𝑤 ∈ 𝐵 ∃𝑦𝜑)
2	1	rexbii 3247	. 2 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∃𝑥∃𝑦𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑥∃𝑤 ∈ 𝐵 ∃𝑦𝜑)
3		rexcom4 3249	. 2 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑥∃𝑤 ∈ 𝐵 ∃𝑦𝜑 ↔ ∃𝑥∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∃𝑦𝜑)
4		rexcom4 3249	. . . . 5 ⊢ (∃𝑤 ∈ 𝐵 ∃𝑦𝜑 ↔ ∃𝑦∃𝑤 ∈ 𝐵 𝜑)
5	4	rexbii 3247	. . . 4 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∃𝑦𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑦∃𝑤 ∈ 𝐵 𝜑)
6		rexcom4 3249	. . . 4 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑦∃𝑤 ∈ 𝐵 𝜑 ↔ ∃𝑦∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜑)
7	5, 6	bitri 277	. . 3 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∃𝑦𝜑 ↔ ∃𝑦∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜑)
8	7	exbii 1848	. 2 ⊢ (∃𝑥∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∃𝑦𝜑 ↔ ∃𝑥∃𝑦∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜑)
9	2, 3, 8	3bitrri 300	1 ⊢ (∃𝑥∃𝑦∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∃𝑥∃𝑦𝜑)

Syntax hints:   ↔ wb 208  ∃wex 1780  ∃wrex 3139

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-11 2161

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-rex 3144

This theorem is referenced by:  satfv1  32610