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Theorem csbeq12 34298
Description: Equality deduction for substitution in class. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
Assertion
Ref Expression
csbeq12 ((𝐴 = 𝐵 ∧ ∀𝑥 𝐶 = 𝐷) → 𝐴 / 𝑥𝐶 = 𝐵 / 𝑥𝐷)

Proof of Theorem csbeq12
StepHypRef Expression
1 csbeq2 3686 . 2 (∀𝑥 𝐶 = 𝐷𝐴 / 𝑥𝐶 = 𝐴 / 𝑥𝐷)
2 csbeq1 3685 . 2 (𝐴 = 𝐵𝐴 / 𝑥𝐷 = 𝐵 / 𝑥𝐷)
31, 2sylan9eqr 2827 1 ((𝐴 = 𝐵 ∧ ∀𝑥 𝐶 = 𝐷) → 𝐴 / 𝑥𝐶 = 𝐵 / 𝑥𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wal 1629   = wceq 1631  csb 3682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-sbc 3588  df-csb 3683
This theorem is referenced by: (None)
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