Theorem cbvral2v 1336

Description: Change bound variables of double restricted universal quantification, using implicit substitution.

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 group(s):   x,z,A   x,y,w,B,z   φ,z   ψ,x,y   χ,x,w

Proof of Theorem cbvral2v

Step	Hyp	Ref	Expression
1		cbvral2v.1	. . . 4 ⊢ (x = z → (φ ↔ χ))
2	1	biraldv 1219	. . 3 ⊢ (x = z → (∀y ∈ B φ ↔ ∀y ∈ B χ))
3	2	cbvralv 1333	. 2 ⊢ (∀x ∈ A ∀y ∈ B φ ↔ ∀z ∈ A ∀y ∈ B χ)
4		cbvral2v.2	. . . 4 ⊢ (y = w → (χ ↔ ψ))
5	4	cbvralv 1333	. . 3 ⊢ (∀y ∈ B χ ↔ ∀w ∈ B ψ)
6	5	biral 1223	. 2 ⊢ (∀z ∈ A ∀y ∈ B χ ↔ ∀z ∈ A ∀w ∈ B ψ)
7	3, 6	bitr 151	1 ⊢ (∀x ∈ A ∀y ∈ B φ ↔ ∀z ∈ A ∀w ∈ B ψ)

Syntax hints:   → wi 2   ↔ wb 127   = weq 797  ∀wral 1201

This theorem is referenced by:  fununi 2705

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-12 802  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-cleq 1097  df-clel 1099  df-ral 1205

metamath.org