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Theorem eqeq12 2107
 Description: Equality relationship among 4 classes. (Contributed by NM, 3-Aug-1994.)
Assertion
Ref Expression
eqeq12 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴 = 𝐶𝐵 = 𝐷))

Proof of Theorem eqeq12
StepHypRef Expression
1 eqeq1 2101 . 2 (𝐴 = 𝐵 → (𝐴 = 𝐶𝐵 = 𝐶))
2 eqeq2 2104 . 2 (𝐶 = 𝐷 → (𝐵 = 𝐶𝐵 = 𝐷))
31, 2sylan9bb 451 1 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴 = 𝐶𝐵 = 𝐷))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1296 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1388  ax-gen 1390  ax-4 1452  ax-17 1471  ax-ext 2077 This theorem depends on definitions:  df-bi 116  df-cleq 2088 This theorem is referenced by:  eqeq12i  2108  eqeq12d  2109  eqeqan12d  2110  funopg  5082  tfri3  6170  th3qlem1  6434  xpdom2  6627  xrlttri3  9366  bcn1  10281  summodc  10926
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