Theorem r2exf 3290

Description: Double restricted existential quantification. For a version based on fewer axioms see r2ex 3205. (Contributed by Mario Carneiro, 14-Oct-2016.) Use r2exlem 3157. (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 3289	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ¬ 𝜑))
3	2	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  Ⅎwnfc 2913  ∃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  ax-6 2000  ax-7 2041  ax-8 2148  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817  df-clel 2841  df-nfc 2915  df-ral 3083  df-rex 3093

This theorem is used by:  rexcomf  3307