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Theorem sbceqi 34224
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 4181 . . 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 2820 . 2 (𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶𝐷 = 𝐸)
73, 6bitri 266 1 ([𝐴 / 𝑥]𝐵 = 𝐶𝐷 = 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wb 197   = wceq 1637  wcel 2156  Vcvv 3391  [wsbc 3633  csb 3728
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-v 3393  df-sbc 3634  df-csb 3729
This theorem is referenced by:  sbccom2lem  34239
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