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) (Proof shortened by Wolf Lammen, 20-Nov-2019)

Ref Expression
Hypotheses eqeqan12d.1 φ A = B
eqeqan12d.2 ψ C = D
Assertion eqeqan12d φ ψ A = C B = D

Proof

Step Hyp Ref Expression
1 eqeqan12d.1 φ A = B
2 eqeqan12d.2 ψ C = D
3 1 adantr φ ψ A = B
4 2 adantl φ ψ C = D
5 3 4 eqeq12d φ ψ A = C B = D