Theorem raltpd 4717

Description: Convert a universal quantification over an unordered triple to a conjunction. (Contributed by Thierry Arnoux, 8-Apr-2019.)

Hypotheses

Ref	Expression
ralprd.1	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
ralprd.2	⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (𝜓 ↔ 𝜃))
raltpd.3	⊢ ((𝜑 ∧ 𝑥 = 𝐶) → (𝜓 ↔ 𝜏))
ralprd.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
ralprd.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)
raltpd.c	⊢ (𝜑 → 𝐶 ∈ 𝑋)

Assertion

Ref	Expression
raltpd	⊢ (𝜑 → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜓 ↔ (𝜒 ∧ 𝜃 ∧ 𝜏)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝜑,𝑥   𝜒,𝑥   𝜃,𝑥   𝜏,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝑉(𝑥)   𝑊(𝑥)   𝑋(𝑥)

Proof of Theorem raltpd

Step	Hyp	Ref	Expression
1		an3andi 1481	. . . . . 6 ⊢ ((𝜑 ∧ (𝜒 ∧ 𝜃 ∧ 𝜏)) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜑 ∧ 𝜃) ∧ (𝜑 ∧ 𝜏)))
2	1	a1i 11	. . . . 5 ⊢ (𝜑 → ((𝜑 ∧ (𝜒 ∧ 𝜃 ∧ 𝜏)) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜑 ∧ 𝜃) ∧ (𝜑 ∧ 𝜏))))
3		ralprd.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4		ralprd.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
5		raltpd.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
6		ralprd.1	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
7	6	expcom 414	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝜑 → (𝜓 ↔ 𝜒)))
8	7	pm5.32d 577	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒)))
9		ralprd.2	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (𝜓 ↔ 𝜃))
10	9	expcom 414	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝜑 → (𝜓 ↔ 𝜃)))
11	10	pm5.32d 577	. . . . . . 7 ⊢ (𝑥 = 𝐵 → ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜃)))
12		raltpd.3	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → (𝜓 ↔ 𝜏))
13	12	expcom 414	. . . . . . . 8 ⊢ (𝑥 = 𝐶 → (𝜑 → (𝜓 ↔ 𝜏)))
14	13	pm5.32d 577	. . . . . . 7 ⊢ (𝑥 = 𝐶 → ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜏)))
15	8, 11, 14	raltpg 4634	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶} (𝜑 ∧ 𝜓) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜑 ∧ 𝜃) ∧ (𝜑 ∧ 𝜏))))
16	3, 4, 5, 15	syl3anc 1370	. . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶} (𝜑 ∧ 𝜓) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜑 ∧ 𝜃) ∧ (𝜑 ∧ 𝜏))))
17	3	tpnzd 4716	. . . . . 6 ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ≠ ∅)
18		r19.28zv 4431	. . . . . 6 ⊢ ({𝐴, 𝐵, 𝐶} ≠ ∅ → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶} (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜓)))
19	17, 18	syl 17	. . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶} (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜓)))
20	2, 16, 19	3bitr2d 307	. . . 4 ⊢ (𝜑 → ((𝜑 ∧ (𝜒 ∧ 𝜃 ∧ 𝜏)) ↔ (𝜑 ∧ ∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜓)))
21	20	bianabs 542	. . 3 ⊢ (𝜑 → ((𝜑 ∧ (𝜒 ∧ 𝜃 ∧ 𝜏)) ↔ ∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜓))
22	21	bicomd 222	. 2 ⊢ (𝜑 → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜓 ↔ (𝜑 ∧ (𝜒 ∧ 𝜃 ∧ 𝜏))))
23	22	bianabs 542	1 ⊢ (𝜑 → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜓 ↔ (𝜒 ∧ 𝜃 ∧ 𝜏)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∅c0 4256  {ctp 4565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-sn 4562  df-pr 4564  df-tp 4566

This theorem is referenced by:  eqwrds3  14676  trgcgrg  26876  tgcgr4  26892  cplgr3v  27802