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Theorem sbcco2 3426
 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 3406 . 2 ([𝑥 / 𝑦][𝐵 / 𝑥]𝜑[𝑥 / 𝑦][𝐵 / 𝑥]𝜑)
2 nfv 1830 . . 3 𝑦[𝐴 / 𝑥]𝜑
3 sbcco2.1 . . . . 5 (𝑥 = 𝑦𝐴 = 𝐵)
43equcoms 1934 . . . 4 (𝑦 = 𝑥𝐴 = 𝐵)
5 dfsbcq 3404 . . . . 5 (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑[𝐵 / 𝑥]𝜑))
65bicomd 212 . . . 4 (𝐴 = 𝐵 → ([𝐵 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
74, 6syl 17 . . 3 (𝑦 = 𝑥 → ([𝐵 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
82, 7sbie 2396 . 2 ([𝑥 / 𝑦][𝐵 / 𝑥]𝜑[𝐴 / 𝑥]𝜑)
91, 8bitr3i 265 1 ([𝑥 / 𝑦][𝐵 / 𝑥]𝜑[𝐴 / 𝑥]𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475  [wsb 1867  [wsbc 3402 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-sbc 3403 This theorem is referenced by:  tfinds2  6955
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