Description: A useful inference for substituting definitions into an equality. (Contributed by NM, 5-Aug-1993) (Proof shortened by Andrew Salmon, 25-May-2011)
Ref | Expression | ||
---|---|---|---|
Hypotheses | eqeq12d.1 | |- ( ph -> A = B ) |
|
eqeq12d.2 | |- ( ph -> C = D ) |
||
Assertion | eqeq12d | |- ( ph -> ( A = C <-> B = D ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq12d.1 | |- ( ph -> A = B ) |
|
2 | eqeq12d.2 | |- ( ph -> C = D ) |
|
3 | eqeq12 | |- ( ( A = B /\ C = D ) -> ( A = C <-> B = D ) ) |
|
4 | 1 2 3 | syl2anc | |- ( ph -> ( A = C <-> B = D ) ) |