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Theorem sbcnestgf 2891
Description: Nest the composition of two substitutions. (Contributed by Mario Carneiro, 11-Nov-2016.)
Assertion
Ref Expression
sbcnestgf ((A 𝑉 yxφ) → ([A / x][B / y]φ[A / xB / y]φ))

Proof of Theorem sbcnestgf
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 dfsbcq 2760 . . . . 5 (z = A → ([z / x][B / y]φ[A / x][B / y]φ))
2 csbeq1 2849 . . . . . 6 (z = Az / xB = A / xB)
3 dfsbcq 2760 . . . . . 6 (z / xB = A / xB → ([z / xB / y]φ[A / xB / y]φ))
42, 3syl 14 . . . . 5 (z = A → ([z / xB / y]φ[A / xB / y]φ))
51, 4bibi12d 224 . . . 4 (z = A → (([z / x][B / y]φ[z / xB / y]φ) ↔ ([A / x][B / y]φ[A / xB / y]φ)))
65imbi2d 219 . . 3 (z = A → ((yxφ → ([z / x][B / y]φ[z / xB / y]φ)) ↔ (yxφ → ([A / x][B / y]φ[A / xB / y]φ))))
7 vex 2554 . . . . 5 z V
87a1i 9 . . . 4 (yxφz V)
9 csbeq1a 2854 . . . . . 6 (x = zB = z / xB)
10 dfsbcq 2760 . . . . . 6 (B = z / xB → ([B / y]φ[z / xB / y]φ))
119, 10syl 14 . . . . 5 (x = z → ([B / y]φ[z / xB / y]φ))
1211adantl 262 . . . 4 ((yxφ x = z) → ([B / y]φ[z / xB / y]φ))
13 nfnf1 1433 . . . . 5 xxφ
1413nfal 1465 . . . 4 xyxφ
15 nfa1 1431 . . . . 5 yyxφ
16 nfcsb1v 2876 . . . . . 6 xz / xB
1716a1i 9 . . . . 5 (yxφxz / xB)
18 sp 1398 . . . . 5 (yxφ → Ⅎxφ)
1915, 17, 18nfsbcd 2777 . . . 4 (yxφ → Ⅎx[z / xB / y]φ)
208, 12, 14, 19sbciedf 2792 . . 3 (yxφ → ([z / x][B / y]φ[z / xB / y]φ))
216, 20vtoclg 2607 . 2 (A 𝑉 → (yxφ → ([A / x][B / y]φ[A / xB / y]φ)))
2221imp 115 1 ((A 𝑉 yxφ) → ([A / x][B / y]φ[A / xB / y]φ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240   = wceq 1242  wnf 1346   wcel 1390  wnfc 2162  Vcvv 2551  [wsbc 2758  csb 2846
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  df-csb 2847
This theorem is referenced by:  csbnestgf  2892  sbcnestg  2893
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