Theorem reean 3317

Description: Rearrange restricted existential quantifiers. For a version based on fewer axioms see reeanv 3239. (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 1947	. . . 4 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
2		reean.1	. . . 4 ⊢ Ⅎ𝑦𝜑
3	1, 2	nfan 1932	. . 3 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝜑)
4		nfv 1947	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵
5		reean.2	. . . 4 ⊢ Ⅎ𝑥𝜓
6	4, 5	nfan 1932	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐵 ∧ 𝜓)
7	3, 6	eean 2382	. 2 ⊢ (∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝑦 ∈ 𝐵 ∧ 𝜓)) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜓)))
8	7	reeanlem 3238	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401  Ⅎwnf 1816   ∈ wcel 2146  ∃wrex 3091

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-6 2000  ax-7 2041  ax-10 2179  ax-11 2195  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-ral 3082  df-rex 3092

This theorem is used by:  disjrnmpt2  45939