Theorem r2ex 3174

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 3173	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ¬ 𝜑))
2	1	r2exlem 3122	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395  ∃wex 1779   ∈ wcel 2109  ∃wrex 3053

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-ral 3045  df-rex 3054

This theorem is referenced by:  r3ex  3176  reeanlem  3208  elxp2  5662  elinxp  5990  rnoprab2  7495  elrnmpores  7527  oeeu  8567  omxpenlem  9042  axcnre  11117  hash2prb  14437  hashle2prv  14443  pmtrrn2  19390  fsumvma  27124  umgredg  29065  fusgr2wsp2nb  30263  spanuni  31473  5oalem7  31589  3oalem3  31593  trsp2cyc  33080  fmla0xp  35370  elfuns  35903  ellines  36140  dalem20  39687  diblsmopel  41165  iunrelexpuztr  43708  sprssspr  47482  prprelb  47517