Theorem r19.29d2r 3291

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 3216	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒) → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝜓 ∧ ∃𝑦 ∈ 𝐵 𝜒))
4	1, 2, 3	syl2anc 587	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝜓 ∧ ∃𝑦 ∈ 𝐵 𝜒))
5		r19.29 3216	. . 3 ⊢ ((∀𝑦 ∈ 𝐵 𝜓 ∧ ∃𝑦 ∈ 𝐵 𝜒) → ∃𝑦 ∈ 𝐵 (𝜓 ∧ 𝜒))
6	5	reximi 3206	. 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 399  ∀wral 3106  ∃wrex 3107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-ral 3111  df-rex 3112

This theorem is referenced by:  ucnima  22887  tgisline  26421  r19.29ffa  30244  xrofsup  30518  icoreresf  34769  sn-negex12  39553