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Theorem csbeq12 36053
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 3816 . 2 (∀𝑥 𝐶 = 𝐷𝐴 / 𝑥𝐶 = 𝐴 / 𝑥𝐷)
2 csbeq1 3814 . 2 (𝐴 = 𝐵𝐴 / 𝑥𝐷 = 𝐵 / 𝑥𝐷)
31, 2sylan9eqr 2800 1 ((𝐴 = 𝐵 ∧ ∀𝑥 𝐶 = 𝐷) → 𝐴 / 𝑥𝐶 = 𝐵 / 𝑥𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wal 1541   = wceq 1543  csb 3811
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 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-sbc 3695  df-csb 3812
This theorem is referenced by: (None)
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