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Theorem sbceqi 4334
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 4333 . . 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 2756 . 2 (𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶𝐷 = 𝐸)
73, 6bitri 278 1 ([𝐴 / 𝑥]𝐵 = 𝐶𝐷 = 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1543  wcel 2111  Vcvv 3415  [wsbc 3703  csb 3820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2072  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-sbc 3704  df-csb 3821
This theorem is referenced by:  sbccom2lem  36032
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