Theorem r19.29d2r 2499

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 2494	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒) → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝜓 ∧ ∃𝑦 ∈ 𝐵 𝜒))
4	1, 2, 3	syl2anc 403	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝜓 ∧ ∃𝑦 ∈ 𝐵 𝜒))
5		r19.29 2494	. . 3 ⊢ ((∀𝑦 ∈ 𝐵 𝜓 ∧ ∃𝑦 ∈ 𝐵 𝜒) → ∃𝑦 ∈ 𝐵 (𝜓 ∧ 𝜒))
6	5	reximi 2458	. 2 ⊢ (∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝜓 ∧ ∃𝑦 ∈ 𝐵 𝜒) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜓 ∧ 𝜒))
7	4, 6	syl 14	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜓 ∧ 𝜒))

Syntax hints:   → wi 4   ∧ wa 102  ∀wral 2348  ∃wrex 2349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-ral 2353  df-rex 2354

This theorem is referenced by:  r19.29vva  2500  cauappcvgprlemdisj  6841