Theorem rsp2e 3283

Description: Restricted specialization. (Contributed by FL, 4-Jun-2012.) (Proof shortened by Wolf Lammen, 7-Jan-2020.)

Assertion

Ref	Expression
rsp2e	⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)

Proof of Theorem rsp2e

Step	Hyp	Ref	Expression
1		rspe 3255	. . 3 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝜑) → ∃𝑦 ∈ 𝐵 𝜑)
2		rspe 3255	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝜑) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)
3	1, 2	sylan2 604	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)
4	3	3impb 1130	1 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   ∈ 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-3an 1103  df-ex 1803  df-rex 3090

This theorem is referenced by:  pell14qrdich  43458