Theorem eqeq12i 2245

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 2244	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))
4	1, 2, 3	mp2an 426	1 ⊢ (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)

Syntax hints:   ↔ wb 105   = wceq 1397

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 1495  ax-gen 1497  ax-4 1558  ax-17 1574  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-cleq 2224

This theorem is referenced by:  rabbi  2711  sbceqg  3143  preqr2g  3850  preqr2  3852  otth  4334  rncoeq  5006  eqfnov  6127  mpo2eqb  6130  f1o2ndf1  6392  ecopovsym  6799  sq11i  10890  dvmptfsum  15448  issubgr  16107  pwle2  16599