Theorem rexlim2d 46199

Description: Inference removing two restricted quantifiers. Same as rexlimdvv 3221, but with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
rexlim2d.x	⊢ Ⅎ𝑥𝜑
rexlim2d.y	⊢ Ⅎ𝑦𝜑
rexlim2d.3	⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝜓 → 𝜒)))

Assertion

Ref	Expression
rexlim2d	⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 → 𝜒))

Distinct variable groups:   𝑦,𝐴   𝜒,𝑥,𝑦

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

Proof of Theorem rexlim2d

Step	Hyp	Ref	Expression
1		rexlim2d.x	. 2 ⊢ Ⅎ𝑥𝜑
2		nfv 1937	. 2 ⊢ Ⅎ𝑥𝜒
3		rexlim2d.y	. . . . 5 ⊢ Ⅎ𝑦𝜑
4		nfv 1937	. . . . 5 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
5	3, 4	nfan 1922	. . . 4 ⊢ Ⅎ𝑦(𝜑 ∧ 𝑥 ∈ 𝐴)
6		nfv 1937	. . . 4 ⊢ Ⅎ𝑦𝜒
7		rexlim2d.3	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝜓 → 𝜒)))
8	7	expdimp 457	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝐵 → (𝜓 → 𝜒)))
9	5, 6, 8	rexlimd 3272	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑦 ∈ 𝐵 𝜓 → 𝜒))
10	9	ex 417	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (∃𝑦 ∈ 𝐵 𝜓 → 𝜒)))
11	1, 2, 10	rexlimd 3272	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 → 𝜒))

Syntax hints:   → wi 4   ∧ wa 400  Ⅎwnf 1806   ∈ 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  ax-6 1990  ax-7 2031  ax-12 2215

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1566  df-ex 1803  df-nf 1807  df-ral 3080  df-rex 3090

This theorem is referenced by:  fourierdlem48  46726