Theorem reean 3310

Description: Rearrange restricted existential quantifiers. For a version based on fewer axioms see reeanv 3223. (Contributed by NM, 27-Oct-2010.) (Proof shortened by Andrew Salmon, 30-May-2011.)

Hypotheses

Ref	Expression
reean.1	⊢ Ⅎ𝑦𝜑
reean.2	⊢ Ⅎ𝑥𝜓

Assertion

Ref	Expression
reean	⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓))

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

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

Proof of Theorem reean

Step	Hyp	Ref	Expression
1		nfv 1910	. . . 4 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
2		reean.1	. . . 4 ⊢ Ⅎ𝑦𝜑
3	1, 2	nfan 1895	. . 3 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝜑)
4		nfv 1910	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵
5		reean.2	. . . 4 ⊢ Ⅎ𝑥𝜓
6	4, 5	nfan 1895	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐵 ∧ 𝜓)
7	3, 6	eean 2340	. 2 ⊢ (∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝑦 ∈ 𝐵 ∧ 𝜓)) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜓)))
8	7	reeanlem 3222	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓))

Syntax hints:   ↔ wb 205   ∧ wa 395  Ⅎwnf 1778   ∈ wcel 2099  ∃wrex 3067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-10 2130  ax-11 2147  ax-12 2167

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-ral 3059  df-rex 3068

This theorem is referenced by:  disjrnmpt2  44561