New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  cbvrex2v GIF version

Theorem cbvrex2v 2844
 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 2635 . . 3 (x = z → (y B φy B χ))
32cbvrexv 2836 . 2 (x A y B φz A y B χ)
4 cbvrex2v.2 . . . 4 (y = w → (χψ))
54cbvrexv 2836 . . 3 (y B χw B ψ)
65rexbii 2639 . 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  ∃wrex 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 2478  df-ral 2619  df-rex 2620 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator