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Theorem gencbval 2596
 Description: Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.) (Proof rewritten by Jim Kingdon, 20-Jun-2018.)
Hypotheses
Ref Expression
gencbval.1 A V
gencbval.2 (A = y → (φψ))
gencbval.3 (A = y → (χθ))
gencbval.4 (θx(χ A = y))
Assertion
Ref Expression
gencbval (x(χφ) ↔ y(θψ))
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 gencbval
StepHypRef Expression
1 alcom 1364 . 2 (xy(y = A → (θψ)) ↔ yx(y = A → (θψ)))
2 gencbval.1 . . . 4 A V
3 gencbval.3 . . . . . . 7 (A = y → (χθ))
4 gencbval.2 . . . . . . 7 (A = y → (φψ))
53, 4imbi12d 223 . . . . . 6 (A = y → ((χφ) ↔ (θψ)))
65bicomd 129 . . . . 5 (A = y → ((θψ) ↔ (χφ)))
76eqcoms 2040 . . . 4 (y = A → ((θψ) ↔ (χφ)))
82, 7ceqsalv 2578 . . 3 (y(y = A → (θψ)) ↔ (χφ))
98albii 1356 . 2 (xy(y = A → (θψ)) ↔ x(χφ))
10 19.23v 1760 . . . 4 (x(y = A → (θψ)) ↔ (x y = A → (θψ)))
11 gencbval.4 . . . . . . 7 (θx(χ A = y))
12 eqcom 2039 . . . . . . . . . 10 (A = yy = A)
1312biimpi 113 . . . . . . . . 9 (A = yy = A)
1413adantl 262 . . . . . . . 8 ((χ A = y) → y = A)
1514eximi 1488 . . . . . . 7 (x(χ A = y) → x y = A)
1611, 15sylbi 114 . . . . . 6 (θx y = A)
17 pm2.04 76 . . . . . 6 ((x y = A → (θψ)) → (θ → (x y = Aψ)))
1816, 17mpdi 38 . . . . 5 ((x y = A → (θψ)) → (θψ))
19 ax-1 5 . . . . 5 ((θψ) → (x y = A → (θψ)))
2018, 19impbii 117 . . . 4 ((x y = A → (θψ)) ↔ (θψ))
2110, 20bitri 173 . . 3 (x(y = A → (θψ)) ↔ (θψ))
2221albii 1356 . 2 (yx(y = A → (θψ)) ↔ y(θψ))
231, 9, 223bitr3i 199 1 (x(χφ) ↔ y(θψ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240   = wceq 1242  ∃wex 1378   ∈ wcel 1390  Vcvv 2551 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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553 This theorem is referenced by: (None)
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