Theorem r19.29vva 2500

Description: A commonly used pattern based on r19.29 2494, version with two restricted quantifiers. (Contributed by Thierry Arnoux, 26-Nov-2017.)

Hypotheses

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

Assertion

Ref	Expression
r19.29vva	⊢ (𝜑 → 𝜒)

Distinct variable groups:   𝑦,𝐴   𝑥,𝑦,𝜒   𝜑,𝑥,𝑦

Allowed substitution hints:   𝜓(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥,𝑦)

Proof of Theorem r19.29vva

Step	Hyp	Ref	Expression
1		r19.29vva.1	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ 𝜓) → 𝜒)
2	1	ex 113	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝜓 → 𝜒))
3	2	ralrimiva 2434	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ 𝐵 (𝜓 → 𝜒))
4	3	ralrimiva 2434	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝜒))
5		r19.29vva.2	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓)
6	4, 5	r19.29d2r 2499	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ((𝜓 → 𝜒) ∧ 𝜓))
7		pm3.35 339	. . . . 5 ⊢ ((𝜓 ∧ (𝜓 → 𝜒)) → 𝜒)
8	7	ancoms 264	. . . 4 ⊢ (((𝜓 → 𝜒) ∧ 𝜓) → 𝜒)
9	8	rexlimivw 2473	. . 3 ⊢ (∃𝑦 ∈ 𝐵 ((𝜓 → 𝜒) ∧ 𝜓) → 𝜒)
10	9	rexlimivw 2473	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ((𝜓 → 𝜒) ∧ 𝜓) → 𝜒)
11	6, 10	syl 14	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ∧ wa 102   ∈ wcel 1433  ∀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-17 1459  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-ral 2353  df-rex 2354

This theorem is referenced by: (None)