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Theorem sbcco2 2780
 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 / x]φ[A / x]φ)
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 2762 . 2 ([x / y][B / x]φ[x / y][B / x]φ)
2 nfv 1418 . . 3 y[A / x]φ
3 sbcco2.1 . . . . 5 (x = yA = B)
43equcoms 1591 . . . 4 (y = xA = B)
5 dfsbcq 2760 . . . . 5 (A = B → ([A / x]φ[B / x]φ))
65bicomd 129 . . . 4 (A = B → ([B / x]φ[A / x]φ))
74, 6syl 14 . . 3 (y = x → ([B / x]φ[A / x]φ))
82, 7sbie 1671 . 2 ([x / y][B / x]φ[A / x]φ)
91, 8bitr3i 175 1 ([x / y][B / x]φ[A / x]φ)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242  [wsb 1642  [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-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-sbc 2759 This theorem is referenced by: (None)
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