Theorem rexlim2d 39857

Description: Inference removing two restricted quantifiers. Same as rexlimdvv 3037, 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 1843	. 2 ⊢ Ⅎ𝑥𝜒
3		rexlim2d.y	. . . . 5 ⊢ Ⅎ𝑦𝜑
4		nfv 1843	. . . . 5 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
5	3, 4	nfan 1828	. . . 4 ⊢ Ⅎ𝑦(𝜑 ∧ 𝑥 ∈ 𝐴)
6		nfv 1843	. . . 4 ⊢ Ⅎ𝑦𝜒
7		rexlim2d.3	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝜓 → 𝜒)))
8	7	expdimp 453	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝐵 → (𝜓 → 𝜒)))
9	5, 6, 8	rexlimd 3026	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑦 ∈ 𝐵 𝜓 → 𝜒))
10	9	ex 450	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (∃𝑦 ∈ 𝐵 𝜓 → 𝜒)))
11	1, 2, 10	rexlimd 3026	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 → 𝜒))

Syntax hints:   → wi 4   ∧ wa 384  Ⅎwnf 1708   ∈ wcel 1990  ∃wrex 2913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-ral 2917  df-rex 2918

This theorem is referenced by:  fourierdlem48  40371