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Theorem sbceqi 32866
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 3935 . . 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 2623 . 2 (𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶𝐷 = 𝐸)
73, 6bitri 262 1 ([𝐴 / 𝑥]𝐵 = 𝐶𝐷 = 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wb 194   = wceq 1474  wcel 1976  Vcvv 3172  [wsbc 3401  csb 3498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-v 3174  df-sbc 3402  df-csb 3499
This theorem is referenced by:  sbccom2lem  32882
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