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Theorem gencbvex 2573
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 1532 . 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 445 . . . . . 6 (A = y → ((χ φ) ↔ (θ ψ)))
65bicomd 129 . . . . 5 (A = y → ((θ ψ) ↔ (χ φ)))
76eqcoms 2021 . . . 4 (y = A → ((θ ψ) ↔ (χ φ)))
82, 7ceqsexv 2566 . . 3 (y(y = A (θ ψ)) ↔ (χ φ))
98exbii 1474 . 2 (xy(y = A (θ ψ)) ↔ x(χ φ))
10 19.41v 1760 . . . 4 (x(y = A (θ ψ)) ↔ (x y = A (θ ψ)))
11 simpr 103 . . . . 5 ((x y = A (θ ψ)) → (θ ψ))
12 gencbvex.4 . . . . . . . 8 (θx(χ A = y))
13 eqcom 2020 . . . . . . . . . . 11 (A = yy = A)
1413biimpi 113 . . . . . . . . . 10 (A = yy = A)
1514adantl 262 . . . . . . . . 9 ((χ A = y) → y = A)
1615eximi 1469 . . . . . . . 8 (x(χ A = y) → x y = A)
1712, 16sylbi 114 . . . . . . 7 (θx y = A)
1817adantr 261 . . . . . 6 ((θ ψ) → x y = A)
1918ancri 307 . . . . 5 ((θ ψ) → (x y = A (θ ψ)))
2011, 19impbii 117 . . . 4 ((x y = A (θ ψ)) ↔ (θ ψ))
2110, 20bitri 173 . . 3 (x(y = A (θ ψ)) ↔ (θ ψ))
2221exbii 1474 . 2 (yx(y = A (θ ψ)) ↔ y(θ ψ))
231, 9, 223bitr3i 199 1 (x(χ φ) ↔ y(θ ψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1226  wex 1358   wcel 1370  Vcvv 2531
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-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-v 2533
This theorem is referenced by:  gencbvex2  2574
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