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Theorem cbvral2v 2843
 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 → (φχ))
cbvral2v.2 (y = w → (χψ))
Assertion
Ref Expression
cbvral2v (x A y B φz 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
StepHypRef Expression
1 cbvral2v.1 . . . 4 (x = z → (φχ))
21ralbidv 2634 . . 3 (x = z → (y B φy B χ))
32cbvralv 2835 . 2 (x A y B φz A y B χ)
4 cbvral2v.2 . . . 4 (y = w → (χψ))
54cbvralv 2835 . . 3 (y B χw B ψ)
65ralbii 2638 . 2 (z A y B χz A w B ψ)
73, 6bitri 240 1 (x A y B φz A w B ψ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642  ∀wral 2614 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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:  cbvral3v  2845  nnpweq  4523  fununi  5160
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