Theorem eqeq12 2251

Description: Equality relationship among 4 classes. (Contributed by NM, 3-Aug-1994.)

Assertion

Ref	Expression
eqeq12	⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))

Proof of Theorem eqeq12

Step	Hyp	Ref	Expression
1		eqeq1 2245	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐶))
2		eqeq2 2248	. 2 ⊢ (𝐶 = 𝐷 → (𝐵 = 𝐶 ↔ 𝐵 = 𝐷))
3	1, 2	sylan9bb 466	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-4 1563  ax-17 1579  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-cleq 2231

This theorem is used by:  eqeq12i  2252  eqeq12d  2253  eqeqan12d  2254  funopg  5414  riotaeqimp  6066  fvdifsuppst  6488  tfri3  6643  th3qlem1  6916  xpdom2  7135  difinfsnlem  7445  difinfsn  7446  xrlttri3  10215  bcn1  11217  summodc  12174  prodmodc  12369  ringinvnz1ne0  14460  wlkres  16842