Theorem r2ex 2527

Description: Double restricted existential quantification. (Contributed by NM, 11-Nov-1995.)

Assertion

Ref	Expression
r2ex	⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑))

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴

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

Proof of Theorem r2ex

Step	Hyp	Ref	Expression
1		nfcv 2349	. 2 ⊢ Ⅎ𝑦𝐴
2	1	r2exf 2525	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑))

Syntax hints:   ∧ wa 104   ↔ wb 105  ∃wex 1516   ∈ wcel 2177  ∃wrex 2486

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-cleq 2199  df-clel 2202  df-nfc 2338  df-rex 2491

This theorem is referenced by:  reean  2676  rexiunxp  4824  rnoprab2  6036  genprndl  7641  genprndu  7642  genpdisj  7643  prmuloc  7686  mullocpr  7691  axcnre  8001