Theorem rspc3v 2964

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 2634	. . . 4 ⊢ (x = A → (∀z ∈ T φ ↔ ∀z ∈ T χ))
3		rspc3v.2	. . . . 5 ⊢ (y = B → (χ ↔ θ))
4	3	ralbidv 2634	. . . 4 ⊢ (y = B → (∀z ∈ T χ ↔ ∀z ∈ T θ))
5	2, 4	rspc2v 2961	. . 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 2951	. . 3 ⊢ (C ∈ T → (∀z ∈ T θ → ψ))
8	5, 7	sylan9 638	. 2 ⊢ (((A ∈ R ∧ B ∈ S) ∧ C ∈ T) → (∀x ∈ R ∀y ∈ S ∀z ∈ T φ → ψ))
9	8	3impa 1146	1 ⊢ ((A ∈ R ∧ B ∈ S ∧ C ∈ T) → (∀x ∈ R ∀y ∈ S ∀z ∈ T φ → ψ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  ∀wral 2614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-v 2861

This theorem is referenced by:  caovassg  5626  caovdig  5632  caovdirg  5633  trd  5921