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Theorem sbciegft 3076
 Description: Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 3077.) (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
Assertion
Ref Expression
sbciegft ((A V xψ x(x = A → (φψ))) → ([̣A / xφψ))
Distinct variable group:   x,A
Allowed substitution hints:   φ(x)   ψ(x)   V(x)

Proof of Theorem sbciegft
StepHypRef Expression
1 sbc5 3070 . . 3 ([̣A / xφx(x = A φ))
2 bi1 178 . . . . . . . 8 ((φψ) → (φψ))
32imim2i 13 . . . . . . 7 ((x = A → (φψ)) → (x = A → (φψ)))
43imp3a 420 . . . . . 6 ((x = A → (φψ)) → ((x = A φ) → ψ))
54alimi 1559 . . . . 5 (x(x = A → (φψ)) → x((x = A φ) → ψ))
6 19.23t 1800 . . . . . 6 (Ⅎxψ → (x((x = A φ) → ψ) ↔ (x(x = A φ) → ψ)))
76biimpa 470 . . . . 5 ((Ⅎxψ x((x = A φ) → ψ)) → (x(x = A φ) → ψ))
85, 7sylan2 460 . . . 4 ((Ⅎxψ x(x = A → (φψ))) → (x(x = A φ) → ψ))
983adant1 973 . . 3 ((A V xψ x(x = A → (φψ))) → (x(x = A φ) → ψ))
101, 9syl5bi 208 . 2 ((A V xψ x(x = A → (φψ))) → ([̣A / xφψ))
11 bi2 189 . . . . . . . 8 ((φψ) → (ψφ))
1211imim2i 13 . . . . . . 7 ((x = A → (φψ)) → (x = A → (ψφ)))
1312com23 72 . . . . . 6 ((x = A → (φψ)) → (ψ → (x = Aφ)))
1413alimi 1559 . . . . 5 (x(x = A → (φψ)) → x(ψ → (x = Aφ)))
15 19.21t 1795 . . . . . 6 (Ⅎxψ → (x(ψ → (x = Aφ)) ↔ (ψx(x = Aφ))))
1615biimpa 470 . . . . 5 ((Ⅎxψ x(ψ → (x = Aφ))) → (ψx(x = Aφ)))
1714, 16sylan2 460 . . . 4 ((Ⅎxψ x(x = A → (φψ))) → (ψx(x = Aφ)))
18173adant1 973 . . 3 ((A V xψ x(x = A → (φψ))) → (ψx(x = Aφ)))
19 sbc6g 3071 . . . 4 (A V → ([̣A / xφx(x = Aφ)))
20193ad2ant1 976 . . 3 ((A V xψ x(x = A → (φψ))) → ([̣A / xφx(x = Aφ)))
2118, 20sylibrd 225 . 2 ((A V xψ x(x = A → (φψ))) → (ψ → [̣A / xφ))
2210, 21impbid 183 1 ((A V xψ x(x = A → (φψ))) → ([̣A / xφψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  [̣wsbc 3046 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-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-sbc 3047 This theorem is referenced by:  sbciegf  3077  sbciedf  3081
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