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Theorem sbied 2036
 Description: Conversion of implicit substitution to explicit substitution (deduction version of sbie 2038). (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.)
Hypotheses
Ref Expression
sbied.1 xφ
sbied.2 (φ → Ⅎxχ)
sbied.3 (φ → (x = y → (ψχ)))
Assertion
Ref Expression
sbied (φ → ([y / x]ψχ))

Proof of Theorem sbied
StepHypRef Expression
1 sb1 1651 . . . 4 ([y / x]ψx(x = y ψ))
2 sbied.1 . . . . 5 xφ
3 sbied.3 . . . . . . 7 (φ → (x = y → (ψχ)))
4 bi1 178 . . . . . . 7 ((ψχ) → (ψχ))
53, 4syl6 29 . . . . . 6 (φ → (x = y → (ψχ)))
65imp3a 420 . . . . 5 (φ → ((x = y ψ) → χ))
72, 6eximd 1770 . . . 4 (φ → (x(x = y ψ) → xχ))
81, 7syl5 28 . . 3 (φ → ([y / x]ψxχ))
9 sbied.2 . . . 4 (φ → Ⅎxχ)
10919.9d 1782 . . 3 (φ → (xχχ))
118, 10syld 40 . 2 (φ → ([y / x]ψχ))
129nfrd 1763 . . 3 (φ → (χxχ))
13 bi2 189 . . . . . . 7 ((ψχ) → (χψ))
143, 13syl6 29 . . . . . 6 (φ → (x = y → (χψ)))
1514com23 72 . . . . 5 (φ → (χ → (x = yψ)))
162, 15alimd 1764 . . . 4 (φ → (xχx(x = yψ)))
17 sb2 2023 . . . 4 (x(x = yψ) → [y / x]ψ)
1816, 17syl6 29 . . 3 (φ → (xχ → [y / x]ψ))
1912, 18syld 40 . 2 (φ → (χ → [y / x]ψ))
2011, 19impbid 183 1 (φ → ([y / x]ψχ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544  [wsb 1648 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  df-sb 1649 This theorem is referenced by:  sbiedv  2037  sbie  2038  dvelimdf  2082  sbco2  2086
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