Theorem cbvral3v 2845

Description: Change bound variables of triple restricted universal quantification, using implicit substitution. (Contributed by NM, 10-May-2005.)

Hypotheses

Ref	Expression
cbvral3v.1	⊢ (x = w → (φ ↔ χ))
cbvral3v.2	⊢ (y = v → (χ ↔ θ))
cbvral3v.3	⊢ (z = u → (θ ↔ ψ))

Assertion

Ref	Expression
cbvral3v	⊢ (∀x ∈ A ∀y ∈ B ∀z ∈ C φ ↔ ∀w ∈ A ∀v ∈ B ∀u ∈ C ψ)

Distinct variable groups:   φ,w   ψ,z   χ,x   χ,v   y,u,θ   x,A   w,A   x,y,B   y,w,B   v,B   x,z,C,y   z,w,C   z,v,C   u,C

Allowed substitution hints:   φ(x,y,z,v,u)   ψ(x,y,w,v,u)   χ(y,z,w,u)   θ(x,z,w,v)   A(y,z,v,u)   B(z,u)

Proof of Theorem cbvral3v

Step	Hyp	Ref	Expression
1		cbvral3v.1	. . . 4 ⊢ (x = w → (φ ↔ χ))
2	1	2ralbidv 2656	. . 3 ⊢ (x = w → (∀y ∈ B ∀z ∈ C φ ↔ ∀y ∈ B ∀z ∈ C χ))
3	2	cbvralv 2835	. 2 ⊢ (∀x ∈ A ∀y ∈ B ∀z ∈ C φ ↔ ∀w ∈ A ∀y ∈ B ∀z ∈ C χ)
4		cbvral3v.2	. . . 4 ⊢ (y = v → (χ ↔ θ))
5		cbvral3v.3	. . . 4 ⊢ (z = u → (θ ↔ ψ))
6	4, 5	cbvral2v 2843	. . 3 ⊢ (∀y ∈ B ∀z ∈ C χ ↔ ∀v ∈ B ∀u ∈ C ψ)
7	6	ralbii 2638	. 2 ⊢ (∀w ∈ A ∀y ∈ B ∀z ∈ C χ ↔ ∀w ∈ A ∀v ∈ B ∀u ∈ C ψ)
8	3, 7	bitri 240	1 ⊢ (∀x ∈ A ∀y ∈ B ∀z ∈ C φ ↔ ∀w ∈ A ∀v ∈ B ∀u ∈ C ψ)

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642  ∀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-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619

This theorem is referenced by: (None)