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Theorem sbciegft 2787
Description: Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 2788.) (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 / x]φψ))
Distinct variable group:   x,A
Allowed substitution hints:   φ(x)   ψ(x)   𝑉(x)

Proof of Theorem sbciegft
StepHypRef Expression
1 sbc5 2781 . . 3 ([A / x]φx(x = A φ))
2 bi1 111 . . . . . . . 8 ((φψ) → (φψ))
32imim2i 12 . . . . . . 7 ((x = A → (φψ)) → (x = A → (φψ)))
43impd 242 . . . . . 6 ((x = A → (φψ)) → ((x = A φ) → ψ))
54alimi 1341 . . . . 5 (x(x = A → (φψ)) → x((x = A φ) → ψ))
6 19.23t 1564 . . . . . 6 (Ⅎxψ → (x((x = A φ) → ψ) ↔ (x(x = A φ) → ψ)))
76biimpa 280 . . . . 5 ((Ⅎxψ x((x = A φ) → ψ)) → (x(x = A φ) → ψ))
85, 7sylan2 270 . . . 4 ((Ⅎxψ x(x = A → (φψ))) → (x(x = A φ) → ψ))
983adant1 921 . . 3 ((A 𝑉 xψ x(x = A → (φψ))) → (x(x = A φ) → ψ))
101, 9syl5bi 141 . 2 ((A 𝑉 xψ x(x = A → (φψ))) → ([A / x]φψ))
11 bi2 121 . . . . . . . 8 ((φψ) → (ψφ))
1211imim2i 12 . . . . . . 7 ((x = A → (φψ)) → (x = A → (ψφ)))
1312com23 72 . . . . . 6 ((x = A → (φψ)) → (ψ → (x = Aφ)))
1413alimi 1341 . . . . 5 (x(x = A → (φψ)) → x(ψ → (x = Aφ)))
15 19.21t 1471 . . . . . 6 (Ⅎxψ → (x(ψ → (x = Aφ)) ↔ (ψx(x = Aφ))))
1615biimpa 280 . . . . 5 ((Ⅎxψ x(ψ → (x = Aφ))) → (ψx(x = Aφ)))
1714, 16sylan2 270 . . . 4 ((Ⅎxψ x(x = A → (φψ))) → (ψx(x = Aφ)))
18173adant1 921 . . 3 ((A 𝑉 xψ x(x = A → (φψ))) → (ψx(x = Aφ)))
19 sbc6g 2782 . . . 4 (A 𝑉 → ([A / x]φx(x = Aφ)))
20193ad2ant1 924 . . 3 ((A 𝑉 xψ x(x = A → (φψ))) → ([A / x]φx(x = Aφ)))
2118, 20sylibrd 158 . 2 ((A 𝑉 xψ x(x = A → (φψ))) → (ψ[A / x]φ))
2210, 21impbid 120 1 ((A 𝑉 xψ x(x = A → (φψ))) → ([A / x]φψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 884  wal 1240   = wceq 1242  wnf 1346  wex 1378   wcel 1390  [wsbc 2758
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-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759
This theorem is referenced by:  sbciegf  2788  sbciedf  2792
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