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Theorem sbcco2 3069
 Description: A composition law for class substitution. Importantly, x may occur free in the class expression substituted for A. (Contributed by NM, 5-Sep-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Hypothesis
Ref Expression
sbcco2.1 (x = yA = B)
Assertion
Ref Expression
sbcco2 ([̣x / y]̣[̣B / xφ ↔ [̣A / xφ)
Distinct variable groups:   x,y   φ,y   y,A
Allowed substitution hints:   φ(x)   A(x)   B(x,y)

Proof of Theorem sbcco2
StepHypRef Expression
1 sbsbc 3050 . 2 ([x / y][̣B / xφ ↔ [̣x / y]̣[̣B / xφ)
2 nfv 1619 . . 3 yA / xφ
3 sbcco2.1 . . . . 5 (x = yA = B)
43eqcoms 2356 . . . 4 (y = xA = B)
5 dfsbcq 3048 . . . . 5 (A = B → ([̣A / xφ ↔ [̣B / xφ))
65bicomd 192 . . . 4 (A = B → ([̣B / xφ ↔ [̣A / xφ))
74, 6syl 15 . . 3 (y = x → ([̣B / xφ ↔ [̣A / xφ))
82, 7sbie 2038 . 2 ([x / y][̣B / xφ ↔ [̣A / xφ)
91, 8bitr3i 242 1 ([̣x / y]̣[̣B / xφ ↔ [̣A / xφ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642  [wsb 1648  [̣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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-sbc 3047 This theorem is referenced by: (None)
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