Theorem rspc3ev 3640

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 1187	. 2 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ 𝜓) → 𝐴 ∈ 𝑅)
2		simpl2 1188	. 2 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ 𝜓) → 𝐵 ∈ 𝑆)
3		rspc3v.3	. . . 4 ⊢ (𝑧 = 𝐶 → (𝜃 ↔ 𝜓))
4	3	rspcev 3626	. . 3 ⊢ ((𝐶 ∈ 𝑇 ∧ 𝜓) → ∃𝑧 ∈ 𝑇 𝜃)
5	4	3ad2antl3 1183	. 2 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ 𝜓) → ∃𝑧 ∈ 𝑇 𝜃)
6		rspc3v.1	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))
7	6	rexbidv 3300	. . 3 ⊢ (𝑥 = 𝐴 → (∃𝑧 ∈ 𝑇 𝜑 ↔ ∃𝑧 ∈ 𝑇 𝜒))
8		rspc3v.2	. . . 4 ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜃))
9	8	rexbidv 3300	. . 3 ⊢ (𝑦 = 𝐵 → (∃𝑧 ∈ 𝑇 𝜒 ↔ ∃𝑧 ∈ 𝑇 𝜃))
10	7, 9	rspc2ev 3638	. 2 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∃𝑧 ∈ 𝑇 𝜃) → ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 ∃𝑧 ∈ 𝑇 𝜑)
11	1, 2, 5, 10	syl3anc 1367	1 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ 𝜓) → ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 ∃𝑧 ∈ 𝑇 𝜑)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1536   ∈ wcel 2113  ∃wrex 3142

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-3an 1085  df-ex 1780  df-cleq 2817  df-clel 2896  df-ral 3146  df-rex 3147

This theorem is referenced by:  f1dom3el3dif  7030  wrdl3s3  14329  pmltpclem1  24052  axlowdim  26750  axeuclidlem  26751  upgr3v3e3cycl  27962  br8d  30364  tgoldbachgt  31938  2goelgoanfmla1  32675  br8  32996  br6  32997  3dim1lem5  36606  lplni2  36677  3cubes  39293  jm2.27  39611