Theorem r2ex 3205

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 3204	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ¬ 𝜑))
2	1	r2exlem 3157	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 1812   ∈ wcel 2146  ∃wrex 3092

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

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-ral 3083  df-rex 3093

This theorem is used by:  r3ex  3207  reeanlem  3239  elxp2  5690  elinxp  6023  rnoprab2  7529  elrnmpores  7561  oeeu  8598  omxpenlem  9076  axcnre  11167  hash2prb  14529  hashle2prv  14535  pmtrrn2  19561  fsumvma  27414  umgredg  29525  fusgr2wsp2nb  30722  spanuni  31933  5oalem7  32049  3oalem3  32053  trsp2cyc  33474  fmla0xp  35896  elfuns  36426  ellines  36665  dalem20  40508  diblsmopel  41986  iunrelexpuztr  44486  sprssspr  48271  prprelb  48306