Theorem rspc3v 2946

Description: 3-variable restricted specialization, using implicit substitution. (Contributed by NM, 10-May-2005.)

Hypotheses

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

Assertion

Ref	Expression
rspc3v	⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑇 𝜑 → 𝜓))

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 rspc3v

Step	Hyp	Ref	Expression
1		rspc3v.1	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))
2	1	ralbidv 2550	. . . 4 ⊢ (𝑥 = 𝐴 → (∀𝑧 ∈ 𝑇 𝜑 ↔ ∀𝑧 ∈ 𝑇 𝜒))
3		rspc3v.2	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜃))
4	3	ralbidv 2550	. . . 4 ⊢ (𝑦 = 𝐵 → (∀𝑧 ∈ 𝑇 𝜒 ↔ ∀𝑧 ∈ 𝑇 𝜃))
5	2, 4	rspc2v 2943	. . 3 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑇 𝜑 → ∀𝑧 ∈ 𝑇 𝜃))
6		rspc3v.3	. . . 4 ⊢ (𝑧 = 𝐶 → (𝜃 ↔ 𝜓))
7	6	rspcv 2925	. . 3 ⊢ (𝐶 ∈ 𝑇 → (∀𝑧 ∈ 𝑇 𝜃 → 𝜓))
8	5, 7	sylan9 413	. 2 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ 𝐶 ∈ 𝑇) → (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑇 𝜑 → 𝜓))
9	8	3impa 1225	1 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑇 𝜑 → 𝜓))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  ∀wral 2528

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823

This theorem is referenced by:  swopolem  4448  isopolem  6022  isosolem  6024  caovassg  6242  caovcang  6245  caovordig  6249  caovordg  6251  caovdig  6258  caovdirg  6261  caoftrn  6329  sgrpass  13706  rngdi  14222  rngdir  14223  islmodd  14612  rmodislmodlem  14670  rmodislmod  14671  lssclg  14684  assalem  14986  psmettri2  15412  xmettri2  15445