Theorem eqeq12 2242

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 2236	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐶))
2		eqeq2 2239	. 2 ⊢ (𝐶 = 𝐷 → (𝐵 = 𝐶 ↔ 𝐵 = 𝐷))
3	1, 2	sylan9bb 462	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-gen 1495  ax-4 1556  ax-17 1572  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-cleq 2222

This theorem is referenced by:  eqeq12i  2243  eqeq12d  2244  eqeqan12d  2245  funopg  5358  riotaeqimp  5991  tfri3  6528  th3qlem1  6801  xpdom2  7010  difinfsnlem  7289  difinfsn  7290  xrlttri3  10022  bcn1  11010  summodc  11934  prodmodc  12129  ringinvnz1ne0  14052  wlkres  16174