Theorem r2ex 3201

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 3200	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ¬ 𝜑))
2	1	r2exlem 3153	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 2145  ∃wrex 3088

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 3079  df-rex 3089

This theorem is used by:  r3ex  3203  reeanlem  3235  elxp2  5683  elinxp  6016  rnoprab2  7523  elrnmpores  7555  oeeu  8595  omxpenlem  9080  axcnre  11177  hash2prb  14541  hashle2prv  14547  pmtrrn2  19593  fsumvma  27457  umgredg  29603  fusgr2wsp2nb  30822  spanuni  32033  5oalem7  32149  3oalem3  32153  trsp2cyc  33571  fmla0xp  35970  elfuns  36500  ellines  36740  dalem20  40574  diblsmopel  42052  iunrelexpuztr  44567  sprssspr  48389  prprelb  48424