Theorem r2exf 3271

Description: Double restricted existential quantification. For a version based on fewer axioms see r2ex 3187. (Contributed by Mario Carneiro, 14-Oct-2016.) Use r2exlem 3135. (Revised by Wolf Lammen, 10-Jan-2020.)

Hypothesis

Ref	Expression
r2exf.1	⊢ Ⅎ𝑦𝐴

Assertion

Ref	Expression
r2exf	⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem r2exf

Step	Hyp	Ref	Expression
1		r2exf.1	. . 3 ⊢ Ⅎ𝑦𝐴
2	1	r2alf 3270	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ¬ 𝜑))
3	2	r2exlem 3135	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 395  ∃wex 1773   ∈ wcel 2098  Ⅎwnfc 2875  ∃wrex 3062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-12 2163

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-nf 1778  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063

This theorem is referenced by:  rexcomf  3292