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Theorem sbcco2 2902
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 2884 . 2 ([𝑥 / 𝑦][𝐵 / 𝑥]𝜑[𝑥 / 𝑦][𝐵 / 𝑥]𝜑)
2 nfv 1491 . . 3 𝑦[𝐴 / 𝑥]𝜑
3 sbcco2.1 . . . . 5 (𝑥 = 𝑦𝐴 = 𝐵)
43equcoms 1667 . . . 4 (𝑦 = 𝑥𝐴 = 𝐵)
5 dfsbcq 2882 . . . . 5 (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑[𝐵 / 𝑥]𝜑))
65bicomd 140 . . . 4 (𝐴 = 𝐵 → ([𝐵 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
74, 6syl 14 . . 3 (𝑦 = 𝑥 → ([𝐵 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
82, 7sbie 1747 . 2 ([𝑥 / 𝑦][𝐵 / 𝑥]𝜑[𝐴 / 𝑥]𝜑)
91, 8bitr3i 185 1 ([𝑥 / 𝑦][𝐵 / 𝑥]𝜑[𝐴 / 𝑥]𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1314  [wsb 1718  [wsbc 2880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-sbc 2881
This theorem is referenced by: (None)
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