Theorem r19.29d2r 3259

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

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		r19.29 3251	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒) → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝜓 ∧ ∃𝑦 ∈ 𝐵 𝜒))
4	1, 2, 3	syl2anc 580	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝜓 ∧ ∃𝑦 ∈ 𝐵 𝜒))
5		r19.29 3251	. . 3 ⊢ ((∀𝑦 ∈ 𝐵 𝜓 ∧ ∃𝑦 ∈ 𝐵 𝜒) → ∃𝑦 ∈ 𝐵 (𝜓 ∧ 𝜒))
6	5	reximi 3189	. 2 ⊢ (∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝜓 ∧ ∃𝑦 ∈ 𝐵 𝜒) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜓 ∧ 𝜒))
7	4, 6	syl 17	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜓 ∧ 𝜒))

Syntax hints:   → wi 4   ∧ wa 385  ∀wral 3087  ∃wrex 3088

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905

This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876  df-ral 3092  df-rex 3093

This theorem is referenced by:  r19.29vva  3260  ucnima  22410  tgisline  25871  r19.29ffa  29837  rnmpt2ss  29983  xrofsup  30043  icoreresf  33690