Metamath Proof Explorer


Theorem eqeqan12d

Description: A useful inference for substituting definitions into an equality. See also eqeqan12dALT . (Contributed by NM, 9-Aug-1994) (Proof shortened by Andrew Salmon, 25-May-2011) Shorten other proofs. (Revised by Wolf Lammen, 23-Oct-2024)

Ref Expression
Hypotheses eqeqan12d.1
|- ( ph -> A = B )
eqeqan12d.2
|- ( ps -> C = D )
Assertion eqeqan12d
|- ( ( ph /\ ps ) -> ( A = C <-> B = D ) )

Proof

Step Hyp Ref Expression
1 eqeqan12d.1
 |-  ( ph -> A = B )
2 eqeqan12d.2
 |-  ( ps -> C = D )
3 1 eqeq1d
 |-  ( ph -> ( A = C <-> B = C ) )
4 2 eqeq2d
 |-  ( ps -> ( B = C <-> B = D ) )
5 3 4 sylan9bb
 |-  ( ( ph /\ ps ) -> ( A = C <-> B = D ) )