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Theorem cbvrex2v 2536
 Description: Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by FL, 2-Jul-2012.)
Hypotheses
Ref Expression
cbvrex2v.1 (x = z → (φχ))
cbvrex2v.2 (y = w → (χψ))
Assertion
Ref Expression
cbvrex2v (x A y B φz A w B ψ)
Distinct variable groups:   x,A   z,A   w,B   x,B,y   z,B,y   χ,w   χ,x   φ,z   ψ,y
Allowed substitution hints:   φ(x,y,w)   ψ(x,z,w)   χ(y,z)   A(y,w)

Proof of Theorem cbvrex2v
StepHypRef Expression
1 cbvrex2v.1 . . . 4 (x = z → (φχ))
21rexbidv 2321 . . 3 (x = z → (y B φy B χ))
32cbvrexv 2528 . 2 (x A y B φz A y B χ)
4 cbvrex2v.2 . . . 4 (y = w → (χψ))
54cbvrexv 2528 . . 3 (y B χw B ψ)
65rexbii 2325 . 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  ∃wrex 2301 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-rex 2306 This theorem is referenced by:  eroveu  6133  genipv  6492
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