Theorem rspc3ev 3623

Description: 3-variable restricted existential specialization, using implicit substitution. (Contributed by NM, 25-Jul-2012.)

Hypotheses

Ref	Expression
rspc3v.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))
rspc3v.2	⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜃))
rspc3v.3	⊢ (𝑧 = 𝐶 → (𝜃 ↔ 𝜓))

Assertion

Ref	Expression
rspc3ev	⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ 𝜓) → ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 ∃𝑧 ∈ 𝑇 𝜑)

Distinct variable groups:   𝜓,𝑧   𝜒,𝑥   𝜃,𝑦   𝑥,𝑦,𝑧,𝐴   𝑦,𝐵,𝑧   𝑧,𝐶   𝑥,𝑅   𝑥,𝑆,𝑦   𝑥,𝑇,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦)   𝜒(𝑦,𝑧)   𝜃(𝑥,𝑧)   𝐵(𝑥)   𝐶(𝑥,𝑦)   𝑅(𝑦,𝑧)   𝑆(𝑧)

Proof of Theorem rspc3ev

Step	Hyp	Ref	Expression
1		simpl1 1192	. 2 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ 𝜓) → 𝐴 ∈ 𝑅)
2		simpl2 1193	. 2 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ 𝜓) → 𝐵 ∈ 𝑆)
3		rspc3v.3	. . . 4 ⊢ (𝑧 = 𝐶 → (𝜃 ↔ 𝜓))
4	3	rspcev 3606	. . 3 ⊢ ((𝐶 ∈ 𝑇 ∧ 𝜓) → ∃𝑧 ∈ 𝑇 𝜃)
5	4	3ad2antl3 1188	. 2 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ 𝜓) → ∃𝑧 ∈ 𝑇 𝜃)
6		rspc3v.1	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))
7	6	rexbidv 3165	. . 3 ⊢ (𝑥 = 𝐴 → (∃𝑧 ∈ 𝑇 𝜑 ↔ ∃𝑧 ∈ 𝑇 𝜒))
8		rspc3v.2	. . . 4 ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜃))
9	8	rexbidv 3165	. . 3 ⊢ (𝑦 = 𝐵 → (∃𝑧 ∈ 𝑇 𝜒 ↔ ∃𝑧 ∈ 𝑇 𝜃))
10	7, 9	rspc2ev 3619	. 2 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∃𝑧 ∈ 𝑇 𝜃) → ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 ∃𝑧 ∈ 𝑇 𝜑)
11	1, 2, 5, 10	syl3anc 1373	1 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ 𝜓) → ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 ∃𝑧 ∈ 𝑇 𝜑)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∃wrex 3061

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062

This theorem is referenced by:  3rspcedvdw  3624  f1dom3el3dif  7267  wrdl3s3  14986  pmltpclem1  25406  axlowdim  28945  axeuclidlem  28946  upgr3v3e3cycl  30166  br8d  32595  tgoldbachgt  34700  2goelgoanfmla1  35451  br8  35778  br6  35779  3dim1lem5  39490  lplni2  39561  3cubes  42680  jm2.27  42999  grimgrtri  47928  usgrexmpl1tri  47996