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Theorem cbvex4v 2012
 Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-Jul-1995.)
Hypotheses
Ref Expression
cbvex4v.1 ((x = v y = u) → (φψ))
cbvex4v.2 ((z = f w = g) → (ψχ))
Assertion
Ref Expression
cbvex4v (xyzwφvufgχ)
Distinct variable groups:   z,w,χ   v,u,φ   x,y,ψ   f,g,ψ   w,f   z,g   w,u,x,y,z,v
Allowed substitution hints:   φ(x,y,z,w,f,g)   ψ(z,w,v,u)   χ(x,y,v,u,f,g)

Proof of Theorem cbvex4v
StepHypRef Expression
1 cbvex4v.1 . . . 4 ((x = v y = u) → (φψ))
212exbidv 1628 . . 3 ((x = v y = u) → (zwφzwψ))
32cbvex2v 2007 . 2 (xyzwφvuzwψ)
4 cbvex4v.2 . . . 4 ((z = f w = g) → (ψχ))
54cbvex2v 2007 . . 3 (zwψfgχ)
652exbii 1583 . 2 (vuzwψvufgχ)
73, 6bitri 240 1 (xyzwφvufgχ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541 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 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545 This theorem is referenced by: (None)
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