Theorem vtocl3ga 2925

Description: Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 20-Aug-1995.)

Hypotheses

Ref	Expression
vtocl3ga.1	⊢ (x = A → (φ ↔ ψ))
vtocl3ga.2	⊢ (y = B → (ψ ↔ χ))
vtocl3ga.3	⊢ (z = C → (χ ↔ θ))
vtocl3ga.4	⊢ ((x ∈ D ∧ y ∈ R ∧ z ∈ S) → φ)

Assertion

Ref	Expression
vtocl3ga	⊢ ((A ∈ D ∧ B ∈ R ∧ C ∈ S) → θ)

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

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

Proof of Theorem vtocl3ga

Step	Hyp	Ref	Expression
1		nfcv 2490	. 2 ⊢ ℲxA
2		nfcv 2490	. 2 ⊢ ℲyA
3		nfcv 2490	. 2 ⊢ ℲzA
4		nfcv 2490	. 2 ⊢ ℲyB
5		nfcv 2490	. 2 ⊢ ℲzB
6		nfcv 2490	. 2 ⊢ ℲzC
7		nfv 1619	. 2 ⊢ Ⅎxψ
8		nfv 1619	. 2 ⊢ Ⅎyχ
9		nfv 1619	. 2 ⊢ Ⅎzθ
10		vtocl3ga.1	. 2 ⊢ (x = A → (φ ↔ ψ))
11		vtocl3ga.2	. 2 ⊢ (y = B → (ψ ↔ χ))
12		vtocl3ga.3	. 2 ⊢ (z = C → (χ ↔ θ))
13		vtocl3ga.4	. 2 ⊢ ((x ∈ D ∧ y ∈ R ∧ z ∈ S) → φ)
14	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13	vtocl3gaf 2924	1 ⊢ ((A ∈ D ∧ B ∈ R ∧ C ∈ S) → θ)

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ w3a 934   = wceq 1642   ∈ wcel 1710

This proof depends on 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 proof 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 2479  df-v 2862

This theorem is used by:  preq12bg  4129