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Theorem sbcco2 3700
 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 3680 . 2 ([𝑥 / 𝑦][𝐵 / 𝑥]𝜑[𝑥 / 𝑦][𝐵 / 𝑥]𝜑)
2 sbcco2.1 . . . . 5 (𝑥 = 𝑦𝐴 = 𝐵)
32equcoms 1978 . . . 4 (𝑦 = 𝑥𝐴 = 𝐵)
4 dfsbcq 3678 . . . . 5 (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑[𝐵 / 𝑥]𝜑))
54bicomd 215 . . . 4 (𝐴 = 𝐵 → ([𝐵 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
63, 5syl 17 . . 3 (𝑦 = 𝑥 → ([𝐵 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
76sbievw 2042 . 2 ([𝑥 / 𝑦][𝐵 / 𝑥]𝜑[𝐴 / 𝑥]𝜑)
81, 7bitr3i 269 1 ([𝑥 / 𝑦][𝐵 / 𝑥]𝜑[𝐴 / 𝑥]𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   = wceq 1508  [wsb 2016  [wsbc 3676 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-ext 2745 This theorem depends on definitions:  df-bi 199  df-an 388  df-ex 1744  df-sb 2017  df-clab 2754  df-cleq 2766  df-clel 2841  df-sbc 3677 This theorem is referenced by:  tfinds2  7393
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