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Theorem cbvral2v 2535
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 2320 . . 3 (x = z → (y B φy B χ))
32cbvralv 2527 . 2 (x A y B φz A y B χ)
4 cbvral2v.2 . . . 4 (y = w → (χψ))
54cbvralv 2527 . . 3 (y B χw B ψ)
65ralbii 2324 . 2 (z A y B χz A w B ψ)
73, 6bitri 173 1 (x A y B φz A w B ψ)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wral 2300
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305
This theorem is referenced by:  cbvral3v  2537  fununi  4908
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