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Theorem sbcco2 2838
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 2820 . 2 ([𝑥 / 𝑦][𝐵 / 𝑥]𝜑[𝑥 / 𝑦][𝐵 / 𝑥]𝜑)
2 nfv 1462 . . 3 𝑦[𝐴 / 𝑥]𝜑
3 sbcco2.1 . . . . 5 (𝑥 = 𝑦𝐴 = 𝐵)
43equcoms 1635 . . . 4 (𝑦 = 𝑥𝐴 = 𝐵)
5 dfsbcq 2818 . . . . 5 (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑[𝐵 / 𝑥]𝜑))
65bicomd 139 . . . 4 (𝐴 = 𝐵 → ([𝐵 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
74, 6syl 14 . . 3 (𝑦 = 𝑥 → ([𝐵 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
82, 7sbie 1715 . 2 ([𝑥 / 𝑦][𝐵 / 𝑥]𝜑[𝐴 / 𝑥]𝜑)
91, 8bitr3i 184 1 ([𝑥 / 𝑦][𝐵 / 𝑥]𝜑[𝐴 / 𝑥]𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1285  [wsb 1686  [wsbc 2816
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-sbc 2817
This theorem is referenced by: (None)
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