Theorem cbvral2v 2844

Description: Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by NM, 10-Aug-2004.)

Hypotheses

Ref	Expression
cbvral2v.1	⊢ (x = z → (φ ↔ χ))
cbvral2v.2	⊢ (y = w → (χ ↔ ψ))

Assertion

Ref	Expression
cbvral2v	⊢ (∀x ∈ A ∀y ∈ B φ ↔ ∀z ∈ A ∀w ∈ B ψ)

Distinct variable groups:   x,A   z,A   x,y,B   y,z,B   w,B   φ,z   ψ,y   χ,x   χ,w

Allowed substitution hints:   φ(x,y,w)   ψ(x,z,w)   χ(y,z)   A(y,w)

Proof of Theorem cbvral2v

Step	Hyp	Ref	Expression
1		cbvral2v.1	. . . 4 ⊢ (x = z → (φ ↔ χ))
2	1	ralbidv 2635	. . 3 ⊢ (x = z → (∀y ∈ B φ ↔ ∀y ∈ B χ))
3	2	cbvralv 2836	. 2 ⊢ (∀x ∈ A ∀y ∈ B φ ↔ ∀z ∈ A ∀y ∈ B χ)
4		cbvral2v.2	. . . 4 ⊢ (y = w → (χ ↔ ψ))
5	4	cbvralv 2836	. . 3 ⊢ (∀y ∈ B χ ↔ ∀w ∈ B ψ)
6	5	ralbii 2639	. 2 ⊢ (∀z ∈ A ∀y ∈ B χ ↔ ∀z ∈ A ∀w ∈ B ψ)
7	3, 6	bitri 240	1 ⊢ (∀x ∈ A ∀y ∈ B φ ↔ ∀z ∈ A ∀w ∈ B ψ)

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642  ∀wral 2615

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 2479  df-ral 2620

This theorem is referenced by:  cbvral3v  2846  nnpweq  4524  fununi  5161