Theorem eqeq12i 2154

Description: A useful inference for substituting definitions into an equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypotheses

Ref	Expression
eqeq12i.1	⊢ 𝐴 = 𝐵
eqeq12i.2	⊢ 𝐶 = 𝐷

Assertion

Ref	Expression
eqeq12i	⊢ (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)

Proof of Theorem eqeq12i

Step	Hyp	Ref	Expression
1		eqeq12i.1	. 2 ⊢ 𝐴 = 𝐵
2		eqeq12i.2	. 2 ⊢ 𝐶 = 𝐷
3		eqeq12 2153	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))
4	1, 2, 3	mp2an 423	1 ⊢ (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)

Syntax hints:   ↔ 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:  rabbi  2611  sbceqg  3023  preqr2g  3702  preqr2  3704  otth  4172  rncoeq  4820  eqfnov  5885  mpo2eqb  5888  f1o2ndf1  6133  ecopovsym  6533  sq11i  10413  pwle2  13366