Theorem r2ex 3202

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 3201	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ¬ 𝜑))
2	1	r2exlem 3154	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑))

Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 400  ∃wex 1802   ∈ wcel 2145  ∃wrex 3089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1803  df-ral 3080  df-rex 3090

This theorem is referenced by:  r3ex  3204  reeanlem  3236  elxp2  5675  elinxp  6008  rnoprab2  7506  elrnmpores  7538  oeeu  8577  omxpenlem  9054  axcnre  11137  hash2prb  14497  hashle2prv  14503  pmtrrn2  19518  fsumvma  27331  umgredg  29393  fusgr2wsp2nb  30590  spanuni  31801  5oalem7  31917  3oalem3  31921  trsp2cyc  33351  fmla0xp  35741  elfuns  36271  ellines  36510  dalem20  40324  diblsmopel  41802  iunrelexpuztr  44302  sprssspr  48086  prprelb  48121