Theorem eqeq12 2153

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 2147	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐶))
2		eqeq2 2150	. 2 ⊢ (𝐶 = 𝐷 → (𝐵 = 𝐶 ↔ 𝐵 = 𝐷))
3	1, 2	sylan9bb 458	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-4 1488  ax-17 1507  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-cleq 2133

This theorem is referenced by:  eqeq12i  2154  eqeq12d  2155  eqeqan12d  2156  funopg  5165  tfri3  6272  th3qlem1  6539  xpdom2  6733  difinfsnlem  6992  difinfsn  6993  xrlttri3  9613  bcn1  10536  summodc  11184  prodmodc  11379