Theorem r2ex 3199

Description: Double restricted existential quantification. (Contributed by NM, 11-Nov-1995.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 10-Jan-2020.)

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		r2al 3198	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ¬ 𝜑))
2	1	r2exlem 3151	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401  ∃wex 1811   ∈ wcel 2144  ∃wrex 3086

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-ral 3077  df-rex 3087

This theorem is used by:  r3ex  3201  reeanlem  3233  elxp2  5674  elinxp  6007  rnoprab2  7526  elrnmpores  7558  oeeu  8615  omxpenlem  9100  axcnre  11252  hash2prb  14620  hashle2prv  14626  pmtrrn2  19679  fsumvma  27545  umgredg  29721  fusgr2wsp2nb  30940  spanuni  32151  5oalem7  32267  3oalem3  32271  trsp2cyc  33730  fmla0xp  36234  elfuns  36769  ellines  37009  dalem20  40843  diblsmopel  42321  iunrelexpuztr  44812  sprssspr  48656  prprelb  48691