Theorem r19.29d2r 3130

Description: Theorem 19.29 of [Margaris] p. 90 with two restricted quantifiers, deduction version. (Contributed by Thierry Arnoux, 30-Jan-2017.) (Proof shortened by Wolf Lammen, 4-Nov-2024.)

Hypotheses

Ref	Expression
r19.29d2r.1	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓)
r19.29d2r.2	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒)

Assertion

Ref	Expression
r19.29d2r	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜓 ∧ 𝜒))

Proof of Theorem r19.29d2r

Step	Hyp	Ref	Expression
1		r19.29d2r.1	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓)
2		r19.29d2r.2	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒)
3	1, 2	jca 510	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒))
4		2r19.29 3129	. 2 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜓 ∧ 𝜒))
5	3, 4	syl 17	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜓 ∧ 𝜒))

Syntax hints:   → wi 4   ∧ wa 394  ∀wral 3051  ∃wrex 3060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804

This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1775  df-ral 3052  df-rex 3061

This theorem is referenced by:  ucnima  24277  tgisline  28554  r19.29ffa  32401  xrofsup  32671  icoreresf  37059  sn-negex12  42196