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Theorem sbceqi 4371
Description: Distribution of class substitution over equality, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)
Hypotheses
Ref Expression
sbceqi.1 𝐴 ∈ V
sbceqi.2 𝐴 / 𝑥𝐵 = 𝐷
sbceqi.3 𝐴 / 𝑥𝐶 = 𝐸
Assertion
Ref Expression
sbceqi ([𝐴 / 𝑥]𝐵 = 𝐶𝐷 = 𝐸)

Proof of Theorem sbceqi
StepHypRef Expression
1 sbceqi.1 . . 3 𝐴 ∈ V
2 sbceqg 4370 . . 3 (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶))
31, 2ax-mp 5 . 2 ([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶)
4 sbceqi.2 . . 3 𝐴 / 𝑥𝐵 = 𝐷
5 sbceqi.3 . . 3 𝐴 / 𝑥𝐶 = 𝐸
64, 5eqeq12i 2751 . 2 (𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶𝐷 = 𝐸)
73, 6bitri 275 1 ([𝐴 / 𝑥]𝐵 = 𝐶𝐷 = 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1542  wcel 2107  Vcvv 3444  [wsbc 3740  csb 3856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-sbc 3741  df-csb 3857
This theorem is referenced by:  sbccom2lem  36629
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