Theorem kmlem10 10048

Description: Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.)

Hypothesis

Ref	Expression
kmlem9.1	⊢ 𝐴 = {𝑢 ∣ ∃𝑡 ∈ 𝑥 𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))}

Assertion

Ref	Expression
kmlem10	⊢ (∀ℎ(∀𝑧 ∈ ℎ ∀𝑤 ∈ ℎ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) → ∃𝑦∀𝑧 ∈ ℎ 𝜑) → ∃𝑦∀𝑧 ∈ 𝐴 𝜑)

Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝑢,𝑡,ℎ   𝑦,𝐴,𝑧,𝑤,ℎ   𝜑,ℎ

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤,𝑢,𝑡)   𝐴(𝑥,𝑢,𝑡)

Proof of Theorem kmlem10

Step	Hyp	Ref	Expression
1		kmlem9.1	. . 3 ⊢ 𝐴 = {𝑢 ∣ ∃𝑡 ∈ 𝑥 𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))}
2	1	kmlem9 10047	. 2 ⊢ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)
3		vex 3440	. . . . 5 ⊢ 𝑥 ∈ V
4	3	abrexex 7894	. . . 4 ⊢ {𝑢 ∣ ∃𝑡 ∈ 𝑥 𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))} ∈ V
5	1, 4	eqeltri 2827	. . 3 ⊢ 𝐴 ∈ V
6		raleq 3289	. . . . 5 ⊢ (ℎ = 𝐴 → (∀𝑤 ∈ ℎ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) ↔ ∀𝑤 ∈ 𝐴 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)))
7	6	raleqbi1dv 3304	. . . 4 ⊢ (ℎ = 𝐴 → (∀𝑧 ∈ ℎ ∀𝑤 ∈ ℎ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)))
8		raleq 3289	. . . . 5 ⊢ (ℎ = 𝐴 → (∀𝑧 ∈ ℎ 𝜑 ↔ ∀𝑧 ∈ 𝐴 𝜑))
9	8	exbidv 1922	. . . 4 ⊢ (ℎ = 𝐴 → (∃𝑦∀𝑧 ∈ ℎ 𝜑 ↔ ∃𝑦∀𝑧 ∈ 𝐴 𝜑))
10	7, 9	imbi12d 344	. . 3 ⊢ (ℎ = 𝐴 → ((∀𝑧 ∈ ℎ ∀𝑤 ∈ ℎ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) → ∃𝑦∀𝑧 ∈ ℎ 𝜑) ↔ (∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) → ∃𝑦∀𝑧 ∈ 𝐴 𝜑)))
11	5, 10	spcv 3560	. 2 ⊢ (∀ℎ(∀𝑧 ∈ ℎ ∀𝑤 ∈ ℎ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) → ∃𝑦∀𝑧 ∈ ℎ 𝜑) → (∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) → ∃𝑦∀𝑧 ∈ 𝐴 𝜑))
12	2, 11	mpi 20	1 ⊢ (∀ℎ(∀𝑧 ∈ ℎ ∀𝑤 ∈ ℎ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) → ∃𝑦∀𝑧 ∈ ℎ 𝜑) → ∃𝑦∀𝑧 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4  ∀wal 1539   = wceq 1541  ∃wex 1780  {cab 2709   ≠ wne 2928  ∀wral 3047  ∃wrex 3056  Vcvv 3436   ∖ cdif 3899   ∩ cin 3901  ∅c0 4283  {csn 4576  ∪ cuni 4859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-rep 5217

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2535  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-in 3909  df-ss 3919  df-nul 4284  df-sn 4577  df-uni 4860

This theorem is referenced by:  kmlem13  10051