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Theorem csbeq12 37620
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 3894 . 2 (∀𝑥 𝐶 = 𝐷𝐴 / 𝑥𝐶 = 𝐴 / 𝑥𝐷)
2 csbeq1 3892 . 2 (𝐴 = 𝐵𝐴 / 𝑥𝐷 = 𝐵 / 𝑥𝐷)
31, 2sylan9eqr 2789 1 ((𝐴 = 𝐵 ∧ ∀𝑥 𝐶 = 𝐷) → 𝐴 / 𝑥𝐶 = 𝐵 / 𝑥𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1532   = wceq 1534  csb 3889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-sbc 3775  df-csb 3890
This theorem is referenced by: (None)
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