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Theorem sbcco 3068
 Description: A composition law for class substitution. (Contributed by NM, 26-Sep-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)
Assertion
Ref Expression
sbcco ([̣A / y]̣[̣y / xφ ↔ [̣A / xφ)
Distinct variable group:   φ,y
Allowed substitution hints:   φ(x)   A(x,y)

Proof of Theorem sbcco
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 sbcex 3055 . 2 ([̣A / y]̣[̣y / xφA V)
2 sbcex 3055 . 2 ([̣A / xφA V)
3 dfsbcq 3048 . . 3 (z = A → ([̣z / y]̣[̣y / xφ ↔ [̣A / y]̣[̣y / xφ))
4 dfsbcq 3048 . . 3 (z = A → ([̣z / xφ ↔ [̣A / xφ))
5 sbsbc 3050 . . . . . 6 ([y / x]φ ↔ [̣y / xφ)
65sbbii 1653 . . . . 5 ([z / y][y / x]φ ↔ [z / y][̣y / xφ)
7 nfv 1619 . . . . . 6 yφ
87sbco2 2086 . . . . 5 ([z / y][y / x]φ ↔ [z / x]φ)
9 sbsbc 3050 . . . . 5 ([z / y][̣y / xφ ↔ [̣z / y]̣[̣y / xφ)
106, 8, 93bitr3ri 267 . . . 4 ([̣z / y]̣[̣y / xφ ↔ [z / x]φ)
11 sbsbc 3050 . . . 4 ([z / x]φ ↔ [̣z / xφ)
1210, 11bitri 240 . . 3 ([̣z / y]̣[̣y / xφ ↔ [̣z / xφ)
133, 4, 12vtoclbg 2915 . 2 (A V → ([̣A / y]̣[̣y / xφ ↔ [̣A / xφ))
141, 2, 13pm5.21nii 342 1 ([̣A / y]̣[̣y / xφ ↔ [̣A / xφ)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176  [wsb 1648   ∈ wcel 1710  Vcvv 2859  [̣wsbc 3046 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-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:  sbc7  3073  sbccom  3117  sbcralt  3118  csbco  3145
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