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Theorem csbeq12 33598
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 3518 . 2 (∀𝑥 𝐶 = 𝐷𝐴 / 𝑥𝐶 = 𝐴 / 𝑥𝐷)
2 csbeq1 3517 . 2 (𝐴 = 𝐵𝐴 / 𝑥𝐷 = 𝐵 / 𝑥𝐷)
31, 2sylan9eqr 2677 1 ((𝐴 = 𝐵 ∧ ∀𝑥 𝐶 = 𝐷) → 𝐴 / 𝑥𝐶 = 𝐵 / 𝑥𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1478   = wceq 1480  csb 3514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-sbc 3418  df-csb 3515
This theorem is referenced by: (None)
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