Theorem r2ex 3190

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 3189	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ¬ 𝜑))
2	1	r2exlem 3138	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 395  ∃wex 1774   ∈ wcel 2099  ∃wrex 3065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775  df-ral 3057  df-rex 3066

This theorem is referenced by:  reeanlem  3220  elxp2  5696  elinxp  6017  rnoprab2  7519  elrnmpores  7553  oeeu  8617  omxpenlem  9089  axcnre  11179  hash2prb  14457  hashle2prv  14463  pmtrrn2  19406  fsumvma  27133  umgredg  28938  fusgr2wsp2nb  30131  spanuni  31341  5oalem7  31457  3oalem3  31461  trsp2cyc  32822  fmla0xp  34929  elfuns  35447  ellines  35684  dalem20  39103  diblsmopel  40581  iunrelexpuztr  43072  sprssspr  46744  prprelb  46779