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Theorem sbcnestgf 3183
 Description: Nest the composition of two substitutions. (Contributed by Mario Carneiro, 11-Nov-2016.)
Assertion
Ref Expression
sbcnestgf ((A V yxφ) → ([̣A / x]̣[̣B / yφ ↔ [̣[A / x]B / yφ))

Proof of Theorem sbcnestgf
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 dfsbcq 3048 . . . . 5 (z = A → ([̣z / x]̣[̣B / yφ ↔ [̣A / x]̣[̣B / yφ))
2 csbeq1 3139 . . . . . 6 (z = A[z / x]B = [A / x]B)
3 dfsbcq 3048 . . . . . 6 ([z / x]B = [A / x]B → ([̣[z / x]B / yφ ↔ [̣[A / x]B / yφ))
42, 3syl 15 . . . . 5 (z = A → ([̣[z / x]B / yφ ↔ [̣[A / x]B / yφ))
51, 4bibi12d 312 . . . 4 (z = A → (([̣z / x]̣[̣B / yφ ↔ [̣[z / x]B / yφ) ↔ ([̣A / x]̣[̣B / yφ ↔ [̣[A / x]B / yφ)))
65imbi2d 307 . . 3 (z = A → ((yxφ → ([̣z / x]̣[̣B / yφ ↔ [̣[z / x]B / yφ)) ↔ (yxφ → ([̣A / x]̣[̣B / yφ ↔ [̣[A / x]B / yφ))))
7 vex 2862 . . . . 5 z V
87a1i 10 . . . 4 (yxφz V)
9 csbeq1a 3144 . . . . . 6 (x = zB = [z / x]B)
10 dfsbcq 3048 . . . . . 6 (B = [z / x]B → ([̣B / yφ ↔ [̣[z / x]B / yφ))
119, 10syl 15 . . . . 5 (x = z → ([̣B / yφ ↔ [̣[z / x]B / yφ))
1211adantl 452 . . . 4 ((yxφ x = z) → ([̣B / yφ ↔ [̣[z / x]B / yφ))
13 nfnf1 1790 . . . . 5 xxφ
1413nfal 1842 . . . 4 xyxφ
15 nfa1 1788 . . . . 5 yyxφ
16 nfcsb1v 3168 . . . . . 6 x[z / x]B
1716a1i 10 . . . . 5 (yxφx[z / x]B)
18 sp 1747 . . . . 5 (yxφ → Ⅎxφ)
1915, 17, 18nfsbcd 3066 . . . 4 (yxφ → Ⅎx[z / x]B / yφ)
208, 12, 14, 19sbciedf 3081 . . 3 (yxφ → ([̣z / x]̣[̣B / yφ ↔ [̣[z / x]B / yφ))
216, 20vtoclg 2914 . 2 (A V → (yxφ → ([̣A / x]̣[̣B / yφ ↔ [̣[A / x]B / yφ)))
2221imp 418 1 ((A V yxφ) → ([̣A / x]̣[̣B / yφ ↔ [̣[A / x]B / yφ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2476  Vcvv 2859  [̣wsbc 3046  [csb 3136 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  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  df-csb 3137 This theorem is referenced by:  csbnestgf  3184  sbcnestg  3185
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