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

Theorem gencbvex 2901
 Description: Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Hypotheses
Ref Expression
gencbvex.1 A V
gencbvex.2 (A = y → (φψ))
gencbvex.3 (A = y → (χθ))
gencbvex.4 (θx(χ A = y))
Assertion
Ref Expression
gencbvex (x(χ φ) ↔ y(θ ψ))
Distinct variable groups:   ψ,x   φ,y   θ,x   χ,y   y,A
Allowed substitution hints:   φ(x)   ψ(y)   χ(x)   θ(y)   A(x)

Proof of Theorem gencbvex
StepHypRef Expression
1 excom 1741 . 2 (xy(y = A (θ ψ)) ↔ yx(y = A (θ ψ)))
2 gencbvex.1 . . . 4 A V
3 gencbvex.3 . . . . . . 7 (A = y → (χθ))
4 gencbvex.2 . . . . . . 7 (A = y → (φψ))
53, 4anbi12d 691 . . . . . 6 (A = y → ((χ φ) ↔ (θ ψ)))
65bicomd 192 . . . . 5 (A = y → ((θ ψ) ↔ (χ φ)))
76eqcoms 2356 . . . 4 (y = A → ((θ ψ) ↔ (χ φ)))
82, 7ceqsexv 2894 . . 3 (y(y = A (θ ψ)) ↔ (χ φ))
98exbii 1582 . 2 (xy(y = A (θ ψ)) ↔ x(χ φ))
10 19.41v 1901 . . . 4 (x(y = A (θ ψ)) ↔ (x y = A (θ ψ)))
11 simpr 447 . . . . 5 ((x y = A (θ ψ)) → (θ ψ))
12 gencbvex.4 . . . . . . . 8 (θx(χ A = y))
13 eqcom 2355 . . . . . . . . . . 11 (A = yy = A)
1413biimpi 186 . . . . . . . . . 10 (A = yy = A)
1514adantl 452 . . . . . . . . 9 ((χ A = y) → y = A)
1615eximi 1576 . . . . . . . 8 (x(χ A = y) → x y = A)
1712, 16sylbi 187 . . . . . . 7 (θx y = A)
1817adantr 451 . . . . . 6 ((θ ψ) → x y = A)
1918ancri 535 . . . . 5 ((θ ψ) → (x y = A (θ ψ)))
2011, 19impbii 180 . . . 4 ((x y = A (θ ψ)) ↔ (θ ψ))
2110, 20bitri 240 . . 3 (x(y = A (θ ψ)) ↔ (θ ψ))
2221exbii 1582 . 2 (yx(y = A (θ ψ)) ↔ y(θ ψ))
231, 9, 223bitr3i 266 1 (x(χ φ) ↔ y(θ ψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859 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-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2861 This theorem is referenced by:  gencbvex2  2902  gencbval  2903
 Copyright terms: Public domain W3C validator