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Theorem csbeq12 33136
 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 3503 . 2 (∀𝑥 𝐶 = 𝐷𝐴 / 𝑥𝐶 = 𝐴 / 𝑥𝐷)
2 csbeq1 3502 . 2 (𝐴 = 𝐵𝐴 / 𝑥𝐷 = 𝐵 / 𝑥𝐷)
31, 2sylan9eqr 2666 1 ((𝐴 = 𝐵 ∧ ∀𝑥 𝐶 = 𝐷) → 𝐴 / 𝑥𝐶 = 𝐵 / 𝑥𝐷)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473   = wceq 1475  ⦋csb 3499 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-sbc 3403  df-csb 3500 This theorem is referenced by: (None)
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