Theorem raltpd 4716

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 1491	. . . . . 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 415	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝜑 → (𝜓 ↔ 𝜒)))
8	7	pm5.32d 583	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒)))
9		ralprd.2	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (𝜓 ↔ 𝜃))
10	9	expcom 415	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝜑 → (𝜓 ↔ 𝜃)))
11	10	pm5.32d 583	. . . . . . 7 ⊢ (𝑥 = 𝐵 → ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜃)))
12		raltpd.3	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → (𝜓 ↔ 𝜏))
13	12	expcom 415	. . . . . . . 8 ⊢ (𝑥 = 𝐶 → (𝜑 → (𝜓 ↔ 𝜏)))
14	13	pm5.32d 583	. . . . . . 7 ⊢ (𝑥 = 𝐶 → ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜏)))
15	8, 11, 14	raltpg 4633	. . . . . 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 1380	. . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶} (𝜑 ∧ 𝜓) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜑 ∧ 𝜃) ∧ (𝜑 ∧ 𝜏))))
17	3	tpnzd 4715	. . . . . 6 ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ≠ ∅)
18		r19.28zv 4437	. . . . . 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 309	. . . 4 ⊢ (𝜑 → ((𝜑 ∧ (𝜒 ∧ 𝜃 ∧ 𝜏)) ↔ (𝜑 ∧ ∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜓)))
21	20	bianabs 547	. . 3 ⊢ (𝜑 → ((𝜑 ∧ (𝜒 ∧ 𝜃 ∧ 𝜏)) ↔ ∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜓))
22	21	bicomd 225	. 2 ⊢ (𝜑 → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜓 ↔ (𝜑 ∧ (𝜒 ∧ 𝜃 ∧ 𝜏))))
23	22	bianabs 547	1 ⊢ (𝜑 → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜓 ↔ (𝜒 ∧ 𝜃 ∧ 𝜏)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121   ≠ wne 2936  ∀wral 3055  ∅c0 4264  {ctp 4562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-12 2191  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-ral 3056  df-rex 3066  df-v 3435  df-dif 3888  df-un 3890  df-nul 4265  df-sn 4559  df-pr 4561  df-tp 4563

This theorem is referenced by:  eqwrds3  14918  trgcgrg  28605  tgcgr4  28621  cplgr3v  29526