Theorem rspc3ev 2904

Description: 3-variable restricted existentional 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 1005	. 2 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ 𝜓) → 𝐴 ∈ 𝑅)
2		simpl2 1006	. 2 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ 𝜓) → 𝐵 ∈ 𝑆)
3		rspc3v.3	. . . 4 ⊢ (𝑧 = 𝐶 → (𝜃 ↔ 𝜓))
4	3	rspcev 2887	. . 3 ⊢ ((𝐶 ∈ 𝑇 ∧ 𝜓) → ∃𝑧 ∈ 𝑇 𝜃)
5	4	3ad2antl3 1166	. 2 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ 𝜓) → ∃𝑧 ∈ 𝑇 𝜃)
6		rspc3v.1	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))
7	6	rexbidv 2511	. . 3 ⊢ (𝑥 = 𝐴 → (∃𝑧 ∈ 𝑇 𝜑 ↔ ∃𝑧 ∈ 𝑇 𝜒))
8		rspc3v.2	. . . 4 ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜃))
9	8	rexbidv 2511	. . 3 ⊢ (𝑦 = 𝐵 → (∃𝑧 ∈ 𝑇 𝜒 ↔ ∃𝑧 ∈ 𝑇 𝜃))
10	7, 9	rspc2ev 2902	. 2 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∃𝑧 ∈ 𝑇 𝜃) → ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 ∃𝑧 ∈ 𝑇 𝜑)
11	1, 2, 5, 10	syl3anc 1252	1 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ 𝜓) → ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 ∃𝑧 ∈ 𝑇 𝜑)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 983   = wceq 1375   ∈ wcel 2180  ∃wrex 2489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-rex 2494  df-v 2781

This theorem is referenced by: (None)